Statistical physics of liquid brains

Abstract

Liquid neural networks (or ‘liquid brains’) are a widespread class of cognitive living networks characterized by a common feature: the agents (ants or immune cells, for example) move in space. Thus, no fixed, long-term agent-agent connections are maintained, in contrast with standard neural systems. How is this class of systems capable of displaying cognitive abilities, from learning to decision-making? In this paper, the collective dynamics, memory and learning properties of liquid brains is explored under the perspective of statistical physics. Using a comparative approach, we review the generic properties of three large classes of systems, namely: standard neural networks (solid brains), ant colonies and the immune system. It is shown that, despite their intrinsic physical differences, these systems share key properties with standard neural systems in terms of formal descriptions, but strongly depart in other ways. On one hand, the attractors found in liquid brains are not always based on connection weights but instead on population abundances. However, some liquid systems use fluctuations in ways similar to those found in cortical networks, suggesting a relevant role for criticality as a way of rapidly reacting to external signals.

This article is part of the theme issue ‘Liquid brains, solid brains: How distributed cognitive architectures process information’.

1. Introduction

As pointed out by physicist John Hopfield, biology is different from physics in one fundamental way: biological systems perform computations [1]. Within the context of evolution, a crucial ingredient for the emergence of biological complexity required the development of information-processing systems at multiple scales [2]. Adaptation to a dynamic environment deeply benefited from non-genetic processes that allowed response mechanisms to short-term changes. Thus, biological computation is an intrinsic part of our current understanding of cell phenotypes [3] and, not surprisingly, the molecular webs of interactions connecting genes, proteins and metabolites have often been represented in terms of computations [4].

Once fast-responding molecular signalling mechanisms were in place, a whole range of possibilities became available: individuals could not only respond to environmental cues, but they could also start to interact with other individuals prompting a higher-order cognitive network [5]. Such transition took place in a diverse range of ways. It included the development of the first brain-like structures [6,7] as well as societies formed by relatively simple agents (ants, termites or bees) capable of performing complex cognitive actions at the collective level [8,9]. Ant colonies have been compared to brains as both exhibit emergent collective phenomena (dynamical and structural patterns of organization and behaviour that cannot be reduced to the properties of single ants) and display cognition on a large scale beyond that of the individual components [10,11]. These two examples represent two distinguishable large classes of networks. Along with ant colonies, immune systems (ISs) also share traits characteristic of the metazoan nerve nets yet they strongly depart from them in the fluid embodiment and nature of cell–cell interactions.

The previous three examples are displayed in figure 1. Here coupled neurons (figure 1a), interacting ants (figure 1b) or immune cells responding to novel challenges (figure 1c) are shown, along with minimal representations of the underlying networks (figure 1d–f). The classical picture of a neural network involves a topological structure (a graph) with neurons occupying the nodes and interneuronal links becoming the edges (figure 1d). Two types of nodes are shown, open and closed, associated to inactive or active neurons, respectively. This is akin to simple interactive models between active (on) and inactive (off) neurons. In this approximation, one treats these agents as small magnets the orientation of which depends on their corresponding neighbours via some non-trivial interaction rules. In general, physicists dub these kinds of systems a spin-glass [12]. This concept lies at the core of what follows here.

Figure 1.

Network interactions in liquid versus solid brains. The three case studies analysed in this paper are shown, with examples of the agents involved in each case. (a) Standard neural networks involve spatially localized cells connected through synaptic weights. In contrast with this architecture, liquid brains, including (b) ant colonies and (c) the IS include mobile agents (or cell subsets) interacting in space and time with no fixed pairwise weights. A schematic of each case study is outlined in the row below. Standard neural networks are defined in terms of connected excitable elements that can be roughly classified in active (firing) and inactive (quiescent) neurons, here indicated as filled and open circles, respectively (d). The wiring matrix remains basically the same in terms of topology (who is connected with whom) but will be modified in strength due to experience. By contrast, ant colonies must be represented by disconnected graphs (e) where interactions are possible within a given spatial range, here indicated by means of the grey circle. The IS allows several representations of the interactions, but in many cases it is the molecular interaction between epitopes (strings of symbols in (f)) that truly represents the underlying liquid brain dynamics.

Ant colonies also involve collectives of interacting individuals that can be modeled as on/off activity - e.g. engaged in a task or not. However, the physical location of such mobile agents change over time: the colony is 'liquid'. Thus, interactions are now extended to a spatiotemporal embedding, with agents interacting at a certain location in one instant and then moving on to other locations to continue interacting with each other. This particularity will constrain the system's adaptive properties in a non-trivial manner. Within the liquid realm, we can still characterize two paradigms: given their relative mobility and signal transmissivity (see below) insect colonies are strongly affected by the locality of their interactions, whereas ISs are highly mobile, such that a well-mixed approach might accurately represent their overall dynamics.

Other types of organisms, such as the slime mould Physarum, solve some classes of optimization problems by using a different form of fluid organization [13], although in this case there is no neural substrate. Systems of this kind are able to solve minimization problems on a network [14].

Upon the transition to multicellularity, cell types capable of sensing and responding to signals appeared and permitted the emergence of a novel class of systems: webs of connected cells. These expanded the landscape of computations, including processing the information in non-trivial (aneural) ways [2]. Such primitive networks provided a reliable way of dealing with complex decisions, integrating and storing memory and creating the conditions for increasing behavioural complexity. Simple organisms such as hydra and planarian flatworms provide good illustrations of the early steps in this direction [7]. To some extent, all these systems can be modelled as networks of neurons that are connected in a stable way over time. Each pair of connected cells will remain linked over a given time scale and changes will take place at the level of the type and strength of the connection. Theoretical work has shown that cognitive tasks performed by these ‘solid’ brains (simple and complex) such as pattern recognition, associative memory or language processing can be properly described. But what about liquid brains?

In this paper, we review several models of both ant colony and IS dynamics based on a neural network perspective and compare them with previous studies on ‘solid’ brain models. In table 1, we summarize some general qualitative properties of the three classes of systems explored here, as well as others that we found relevant. The list is not exhaustive and involves generic descriptors that inevitably ignore the broad diversity of sizes, organization levels and ecological contexts. Several key components of each potential candidate, including size, age, context or developmental trajectories have some influence in the degree of robustness, memory potential or wiring patterns. All these factors make this basic table a tentative one. Nevertheless, it also highlights the commonalities that we consider relevant to our presentation.

Table 1.

Comparative properties in liquid versus solid brains. This table summarizes a broad set of properties that are usually attributed to neural systems (solid brains) and here compared to those reported from two relevant examples of liquid brains, namely the IS and insect (mostly ant) colonies. While the way computations are performed is a parallel process in all systems, all also exhibit some degree of specialization, which can be understood as a modularity or a division of labour (DOL). This first is observable in vertebrate brains while the later is a characteristic allocation of tasks that can occur either in societies with different morphological castes or in monomorphic ones. Similarly, we label the learning and memory properties in terms of a simple, network-related set of properties. In most cases studied here, the memory potential of an ant colony is related to short-term phenomena tied to the production of a pheromone field, but long-term memories have also been reported at the individual level. In all these examples, we indicate by asterisk (*) those attributes that are not well established or have been found in some case studies, and that will benefit from a theory of liquid brains.

	brain	insect colonies	IS
computational nature	distributed/modular	distributed/DOL	distributed/DOL
reliability under agent loss	high	high	high
connectivity	hard-wired yet plastic	liquid but spatially constrained (*)	liquid (*)
memory and learning	synaptic	population-based and synaptic	population-based (Burnet) and synaptic (Jerne)
regeneration potential	low	high	high
externalities	peripheral nervous system and technology (tools)	nests and agriculture	no
weighted interactions	Hebbian	pheromone-mediated	antibody-mediated
dynamical state	critical	critical (*)	critical (*)

Some key examples are worth mentioning. The label ‘liquid’ is used to describe a physical state that ignores spatial structuring, such as lymph nodes in the IS or the nest structure of ant or termite colonies. Some of these features cannot be taken as absolute indicators since they are strongly influenced by life styles, size or behavioural context. The neural network of a hydra or a planarian flatworm are simple and small and might not display the modularity found in more complex neural agents, but nevertheless they display spatially stable networks of neurons, which are reliable under cell loss. Other relevant features (which are not included in table 1) such as the self/non-self discrimination problem will be amply discussed later on. In the following sections, we summarize several types of models used to represent and understand the dynamics of the three case studies discussed here. By using them, we aim at enhancing the universal elements shared by these liquid systems while tracing a theoretical framework to study them.

2. Solid brains

Standard neural networks (NN), from cell cultures to brains, have received great attention since the 1950s. An especially successful approach has been based on the use of statistical physics as a robust formalism capable of capturing the collective properties exhibited by neural masses [15]. Both in statistical physics as well as in logic models of NN, neurons are replaced by a toy model representing only the minimal features exhibited by real cells. The intricate structure of physiological neurons is ignored and replaced by a formal object devoid of any specific traits associated to cellular or molecular biological mechanisms. Similarly, the way connections and propagation of activity occurs is mapped into a simple graph. Despite all these oversimplifications, NN theory (also known as connectionism) has been capable of explaining the nature and relevance of collective phenomena involved in a broad range of areas, from learning in small metazoans to more complex phenomena related to human cognition [16,17].

We use here the term ‘brains’ in a generic way too: it will refer to ensembles of interconnected neurons (or neural-like elements). Over the years the field has been growing in multiple directions, but a special turning point is the classical paper by Hopfield [18] where the basis for a statistical physics description of neural networks emerged and largely marked the development of this class of systems. Such a physics perspective provided the basis for the understanding of their global properties out from the underlying microscopic description. Importantly, it also provided a systematic approach to identify the presence of different phases associated with the presence or lack of memory as well as dynamical states separating different types of activity. In this way, the physics of phase transitions [19–21] became a cornerstone to our understanding of neural networks.

The simplest, canonical model is based on an assembly of two-state agents description [22,23]. These are denoted as S_i(t) ∈ {0, 1} or S_i(t) ∈ {−1, 1} (with i = 1, …, N). Agents are connected to each other through fixed synaptic links (figure 1a): each element sends to and receives a signal from another. Connectivity is represented by a matrix J_ik ∈ R. The system is modelled by a dynamical set of equations:

$$S_i(t+1)=\mathit{\Theta }\hspace{0.17em}(\sum _{\hspace{0.17em}j=1}^N{J}_{ij}{S}_{\hspace{0.17em}j}(t)-{\theta }_i),$$ 2.1

where Θ (z) = 1 for z > 0 and zero otherwise. The scalar θ_i is a threshold value. The so called external field, $h_i=\sum _{\hspace{0.17em}j}{J}_{ij}{S}_{\hspace{0.17em}j}(t)$, weights the total input of S_i. It is worth noting that the same class of threshold model used to describe the dynamics of NN has been used to approach the dynamics of gene regulatory networks (GRN) [24, pp. 411–521, 25,26].

(a). Attractor dynamics in recurrent neural networks

A general treatment of these systems involves a high-dimensional problem and a wide range of dynamical behaviours. However, an illustration of the potential of NN as a way of solving computational problems in a distributed manner is provided by the Hopfield model [18,27]. This consists of a fully connected neural network described by the dynamical equations (2.1) with θ_i = 0. Hopfield’s model assumes no self-connection (J_ii = 0) and symmetry, i.e. J_ij = J_ji. It can be shown that the model only displays single-point equilibrium (attractors), i.e. asymptotically, the trained network will tend to a stable configuration where all elements remain in a given state (figure 2a–c). Additionally, Hopfield’s model allows the network to store a number p of ‘memories’ (patterns) defined as a set of vectors ${\xi }_{\mu }=({\xi }_1^{\mu },\dots ,{\xi }_{\hspace{0.17em}j}^{\mu },\dots ,{\xi }_N^{\mu })$, μ = 1…, p. The storage process takes place within a ‘training phase’ where they are presented to the network in such a way that each neuron S_i adopts the memory state, i.e. S_i = ξ_i and all its synaptic weights J_ij are updated (starting from J_ij = 0 at time zero) following the so-called Hebb’s rule, which is summarized in figure 2b. In a nutshell, correlated inputs increase weights, whereas uncorrelated ones decrease them. It can be shown [28] that the memory states ξ_μ are, in fact, the minima of a (high-dimensional) energy function, namely:

$$\mathcal{H}(\{{\xi }_i^{\mu }\})=-\frac{1}{2}\sum _{i,j}{J}_{ij}{S}_i{S}_{\hspace{0.17em}j},J_{ij}=\frac{1}{N}\sum _{\mu =0}^{\hspace{0.17em}p}{\xi }_i^{\mu }{\xi }_{\hspace{0.17em}j}^{\mu }$$ 2.2

and initial conditions close to a minimum will evolve towards it. This is also outlined in figure 2c where we represent such multiple minima. In summary, the Hopfield model is a dynamical process of memory retrieval: stored patterns are recovered by a purely dynamical process. Extensions to this approach come by introducing thermal noise for the {S_i} degrees of freedom. Usually, this is obtained via a temperature T that accounts for stochastic thermal variations (and, more generally, for noise). Each time we choose a neuron, the probability of changing to (or remaining in) state S_i = +1 is a saturating function of the local field, namely:

$$P[S_i(t+1)=+1\hspace{0.17em}|\hspace{0.17em}{h}_i(t)]=\varphi (\frac{1}{T}\sum _{\hspace{0.17em}j}{J}_{ij}{S}_{\hspace{0.17em}j}(t))$$ 2.3

with T defining a temperature and ϕ(x) a function such that ϕ(0) = 0 and ϕ(x) → ±1 for x → ±∞. Temperature is not just an additional attribute, as it actually provides a powerful mechanism to escape from local minima. By using a stochastic transition rule, it is possible to move to lower-energy states from a given, suboptimal (usually non-memory) state. In this context, a measure of memory capacity is introduced as α ≡ p/N, where p here corresponds to the number of well-stored patterns. A phase-transition diagram captures the overall system behaviour, depicted in figure 2d. The shaded region represents states where the system is capable of retaining the memory patterns, while, for the blank region, these are lost due to noise. An abrupt transition separates these two regimes.

Figure 2.

Distributed computation in neural networks. Using a very simple set of rules, an NN model can store and retrieve memories in a robust manner. In the Hopfield’s model, a massively connected set of neurons (a) with symmetric connections obeying Hebb’s rule (b) will display such properties. In (b), a pair of formal neurons is shown receiving inputs ξ_i, ξ_j ∈ {−1, 1} from a given memory state or pattern ξ^μ. If they are identical, i.e. ξ_i = ξ_j, their connection is increased (in both directions). Otherwise, J_ij is decreased. (c) Network dynamics makes the system’s state flow to energy minima, thus recovering the desired memory state. The model exhibits remarkable reliability against connection loss. In (d), we show how reliable is memory retrieval against stochastic thermal variability. Parameter α is a relative measure of memory capacity. The critical value α_c ≃ 0.138 separates the two phases: memory reliability (shaded area) and unreliability (blank area). This transition occurs sharply. Note that this critical value is specific for Hopfield nets; different interaction rules would yield different limitations to memory capacity.

The previous model is an illustration of how cognitive functions can be understood in terms of a system of connected neurons. Here synaptic weights are modified in such a way that the resulting attractor dynamics allows associative memory to be the consequence of a relaxation towards energy minima. Only steady states are thus allowed. However, as discussed in the next section, a different picture emerges when we look at the actual dynamical patterns exhibited by neural tissues.

(b). Critical dynamics in cortical networks

If we think of an idealized graph such as the one described in figure 1d, two classes of nodes can be defined: either inactive or active. Active nodes are formed by firing neurons whose excitability can be propagated to the nearest inactive areas [29]. As a result, excitation waves can move across whole areas. This would be a requirement to maintain integration in a dynamical fashion [30]. This analysis gives rise to more complex types of attractors instead of local stable points.

The minimal model that can describe the propagation or activity is based on a contagion scenario where inactive nodes can become active if they are connected to active nodes. Moreover, an active node can spontaneously decay. At the smallest scale, this is similar to the threshold dynamics described above. The simplest case to consider is a homogeneous model where all connections are similar, capable of propagating excitability with J_ij = J and an average connectivity 〈k〉. It can be shown that the large-scale (coarse-grained) dynamics for this homogeneous case can be defined by the equation [29]

$$\frac{\text{d}A}{\text{d}t}=f(A)=-\frac{1}{\tau }A+\frac{J⟨k⟩}{\tau }A(1-A),$$ 2.4

where τ is a characteristic time decay. A specially relevant observation is that neural systems exhibit critical behaviour [31–33]. Two main classes of dynamical behaviour can occur. This can be shown using the fixed points, i.e. those A* such that (dA/dt)_A* = 0. Two states are obtained. One is the trivial, inactive phase where no activity propagates: $A_0^{\ast }=0$. The second phase is associated with the second fixed point, namely:

$$A_1^{\ast }=1-\frac{1}{J⟨k⟩},$$ 2.5

which is properly defined (i.e. $A_1^{\ast }\ge 0$) provided that J〈k〉 ≥ 1. A critical point separating the two phases is thus achieved for J〈k〉 = 1. For a given J value, the critical connectivity is given by 〈k〉_c = 1/J.

In figure 3, two important diagrams are shown that summarize the basic phenomena resulting from the previous model. One is the so-called bifurcation diagram [35] where the stable states $A_0^{\ast }$, $A_1^{\ast }$ are plotted against the average connectivity 〈k〉, with a marked change occurring at criticality (figure 3c). Additionally, we also display (figure 3d) the potential function $V(\mathcal{A})$ [36], defined as

$$V(\mathcal{A})=-\int f(A)\hspace{0.17em}\text{d}\mathcal{A},$$ 2.6

such that the dynamics derive from it, i.e. $\text{d}\mathcal{A}/\text{d}t=-\text{d}V(\mathcal{A})/\text{d}\mathcal{A}$. The minima (maxima) of the potential correspond to stable (unstable) fixed points. As we approach criticality, the potential function becomes increasingly flatter. What is the impact of this flatness in the activity? In general, shallow potentials are associated with higher time variability and fluctuations diverge close to criticality. To show this, we can use a linear stability analysis taking the state $\mathcal{A}(t)={\mathcal{A}}_k^{\ast }+\delta \mathcal{A}$, i.e. a small deviation $\delta \mathcal{A}$ from a fixed point $A_k^{\ast }$, and plugging it into the original equation for $\mathcal{A}(t)$. On a first approximation, it can be shown that

$$\frac{\text{d}\delta \mathcal{A}}{\text{d}t}={\left(\frac{\mathrm{\partial }f(\mathcal{A})}{\mathrm{\partial }\mathcal{A}}\right)}_{A_k^{\ast }}\delta \mathcal{A}={\lambda }_k^{\ast }\delta \mathcal{A},$$ 2.7

where ${\lambda }_k^{\ast }$ is a scalar to be evaluated at each fixed point. The resulting equation for fluctuations is linear. Thus, close to $A_k^{\ast }$, we expect a growth of fluctuations following an exponential growth or decay. For the inactive phase,¹ we have

$$\delta \mathcal{A}(t)={(\delta \mathcal{A})}_0\hspace{0.17em}{\text{e}}^{{\lambda }_0^{\ast }t}={(\delta \mathcal{A})}_0\exp (\frac{-1}{\tau }(1-J⟨k⟩)t).$$ 2.8

As we can see, the system will return to the fixed point (when 〈k〉 < 1/J) at a rate given by λ₀. As we get close to criticality, the exponent gets smaller, the relaxation time rapidly increases. If the previous result is written in terms of a relaxation time T(J, 〈k〉), i.e. $\delta \mathcal{A}(t)\sim \exp (-t/T(J,⟨k⟩))$ we have

$$T(J,⟨k⟩)\sim \frac{1}{1-J⟨k⟩},$$ 2.9

which rapidly diverges as J〈k〉 → 1. The divergence predicted by this simple model is confirmed by the analysis of the fluctuations found in neural systems.

Figure 3.

Phase transitions in neural dynamics. In a simple version of large-scale dynamics of neural tissues, (a) tissues (such as brain cortex) can be represented as a network of connected neighbouring areas that are connected with excitatory links (adapted from Eckmann et al. [34]). A toy model of this (b) could be represented as a lattice of neural elements connected as a grid with all elements linked to four elements in a homogeneous fashion. The analysis of this system reveals a phase transition from zero activity to high-activity by crossing a critical value of average connections at 〈k〉_c = 1/J (c). A potential function can be obtained where the two phases are revealed as stable states of $V(\mathcal{A})$ (d). Here, large fluctuations show clear dominance around the critical point.

The two previous models explore some essential components of neural complexity. Both deal with collective behaviour and exhibit special regions of parameter spaces that separate different phases. Phase transitions are of central importance within statistical physics, and provide a powerful framework to capture how microscopic interactions translate into system-level patterns and processes [36,37]. Their importance within our context becomes manifest as qualitative changes in collective behaviour are typically caused by phase transition phenomena often associated with the density of individuals or the signals they use to communicate. How these systems behave close to transition points turns out to be a key issue, as it provides understanding about how emergent phenomena occur.

3. Liquid brains

(a). Ant colony dynamics

Social insects, including ants and termites among other groups, amount to about the same biomass as humans on the Earth [38]. With an evolutionary history spanning around a hundred million years, eusocial colonies have deeply engineered the environment and dominated the terrestrial biosphere [39]. In trying to attach biological fitness, insect colonies appear to behave as superorganisms: it is the colony as a whole that plays an evolutionary role, rather than its individual agents (ants). Across the biosphere, we encounter both monomorphic and polymorphic ant colonies, the latter exhibiting physiological–anatomical differences within a given colony. However, it is estimated that 80% of ant species are monomorphic. The rest of species (polymorphic) can generally include two or three different casts. Here onwards, we will focus our study on monomorphic ant species.

On the other hand, various estimates state that the behavioural repertoire of ant colonies ranges from 20 to 45 different individual-ant behaviours [8, pp. 180–200]. In order to shift from a given state to another and adapt to any given environmental circumstance, ants use chemical signals called pheromones. Different ant species use different sets of pheromones, some secrete only one type of molecule and others use up to 20.² Thus, information is processed in a two-level fashion: mobile agents (ants) interacting with a set of diffusive molecules (pheromones). Ants continuously detect the pheromone concentrations and, upon integrating this information, produce an internal image that affects their behavioural state. Moreover, ant states prompt the secretion of one (or more) pheromones thus reshaping their concentration values. This coalescence of signalling back and forth allows the whole colony to access global states where functions are achieved by means of its underlying network of interactions. Information is stored and processed through this ‘liquid brain’ to give rise to various large-scale collective behaviours. In the following examples, we will review several theoretical approaches to modelling ant colony dynamics and compare them with standard NN model efforts.

(i). Ant colonies as excitable neural nets

One of the simplest illustrations of the neural-like modelling of insect colony dynamics is provided by the emergent synchronization displayed by some small colonies of the genus Leptothorax ([40,41]; see also [9], ch. 6). In a nutshell, it has been observed that the colony-level activity displayed by their nests exhibits a remarkable bursting pattern (figure 4a,b) that exhibits a periodic component [42]. This means that ants can be active or inactive and the total number of active individuals changes in such a way that at times no ant in the colony is active while the synchronization events are linked to an almost fully active colony. These bursts have been found in other species [43] and result from the propagation of activity carried by moving individuals that can activate dormant ants in ways similar to those found in epidemic models [36,44,45]. Synchronization of neural masses is in fact a major research field within neuroscience [46], and it has been shown to pervade a wide range of functional traits and behavioural patterns. Is there something similar taking place in ant colonies?

Figure 4.

Ant colonies as excitable neural nets. In some ant species, such as those belonging to the genus Leptothorax (a), oscillations in activity have been recorded (b) revealing a collective synchronization phenomenon (both adapted from Solé [36]). This phenomenon can be described as an excitable neural system, where ants (inset of c) are reduced to a Boolean representation with active and inactive individuals. (c) As the density of ants ρ increases, a phase change occurs at a critical density, separating inactive from active colonies. (d) Potential function associated with the dynamics of these colonies: for densities larger (lower) than ρ_c, a well-defined minimum is displayed. Closer to criticality, this potential becomes flatter and allows wide fluctuations to occur.

This problem provides a simple example of a fluid network. Here the description level of individuals and their interactions is limited to a Boolean set of variables Σ = {0, 1} associated with the inactive (motionless) and active (moving) states, respectively (see inset of figure 4c). An NN model here is thus limited to a coarse-grained representation of ants. Such a model was suggested in Solé et al. [40] under the assumption that individuals can be described as an underlying continuous variable S_i ∈ [0, 1] (with i = 1, …, N) which changes over time following a dynamical equation:

$$S_i(t+1)=\mathit{\Theta }\hspace{0.17em}(\sum _{\hspace{0.17em}j\in {\mathit{\Gamma }}_i}J({\eta }_{\hspace{0.17em}j}(t),{\eta }_i(t))S_{\hspace{0.17em}j}(t)),$$ 3.1

which strongly resembles the familiar form of standard NN. However, a rapid inspection reveals a fundamental difference: here the matrix J(η_j, η_i) is state-dependent. In other words, its value is a function of the specific pair of agents that interact at a given time step. Specifically, we partition the activity interval [0, 1] into two domains associated with the active/inactive observables, i.e. η_i = Θ [S_i − θ]. Thus, the interaction matrix will include only four possible pairs,

$$[J({\eta }_{\hspace{0.17em}j},{\eta }_i)]=\left[\begin{array}{cc}{J}_{00}& J_{01}\\ J_{10}& J_{11}\end{array}\right],$$

where J ≥ 0. Once activity decreases below the threshold θ, the ant becomes inactive and stops moving. Otherwise, it moves around as a random walker (unless constrained by other ants occupying the nearest lattice sites). Here ants are assumed to move on a discrete two-dimensional lattice Ω and interactions occur in a strictly local manner, only affecting the set of nearest neighbouring positions Γ_i of S_i. Finally, an inactive ant (with η < θ) can become active spontaneously (achieving a state S₀ > θ) with probability p_a. A common feature of these matrices is the presence of coupling terms connecting active and inactive individuals, as expected from an excitable system where activity can be propagated among agents. It is important to notice that the collective synchronization does not result from the coupling of individuals’ internal clock. Instead, single virtual ants behave randomly. The dynamics of single elements will be described by

$$S_i(t+1)=\mathit{\Theta }\hspace{0.17em}(g{J}_{11}{S}_{\hspace{0.17em}j}(t)).$$ 3.2

A simple case can be solved, namely when the coupling is small and activity remains small (which is consistent with observation). If we choose Θ (x) = tanh x, then we may use linear approximation tanh (gJz) ≈ gJz which admits a solution to the previous equation. If, initially, an ant is activated to a level S₀, then S(t) = S₀(gJ)^t, which is a decaying function of time. If an activation term is also introduced (i.e. active ants can activate inactive ones), then a coarse-grained model can be defined in probabilistic terms. Let us label as N_a the number of active ants. This number will change over time as a consequence of both interactions and decay. The efficiency of activation events will be proportional to gJ, assuming the previous linear approximation. Hereafter, we will indicate by N and ρ the total number and density of ants, respectively.

If $\mathcal{A}(\mathbf{x},t)$ indicates the probability density of active ants at a given point of our two-dimensional lattice x ∈ Ω, then it can be written as: $\mathcal{A}(\mathbf{x},t)=P[S_{\mathbf{x}}(t)=1]$. The activity density will evolve following a master equation according to the previous rules:

$$\frac{\text{d}\mathcal{A}(\mathbf{x},t)}{\text{d}t}=\frac{gJ}{q}\sum _{⟨\mathbf{u}⟩}^{q}P[S_{\mathbf{x}}(t)=0\cap S_{\mathbf{u}}(t)=1]-\alpha \mathcal{A}(\mathbf{x},t),$$ 3.3

where 〈u〉 indicates sum over the set of q nearest neighbours, $P[S_{\mathbf{x}}=0\cap S_{\mathbf{u}}=1]$ is the probability of having a pair of nearest ants in different states.

The previous equation is exact, but its computation would require knowledge of the probabilities associated with the interactions between nearest sites. Several methods can be used to solve this model with different levels of approximation. Here we will consider the simplest one, commonly known as a mean field theory, which is based on suppressing the spatial correlation between nearest sites. This is done by assuming that the system is in fact well mixed and thus all sites are neighbours or, in mathematical terms, q = Ω. If this is the case, we can use the total population

$$\rho (t)=\sum _{\mathbf{x}}^{\mathit{\Omega }}\rho (\mathbf{x},t).$$ 3.4

By summing on both sides of the previous master equation, and ignoring correlations between active and inactive neighbours, it can be shown that the global dynamics can be described as

$$\frac{\text{d}\mathcal{A}}{\text{d}t}=gJ\mathcal{A}(\rho -\mathcal{A})-\alpha \mathcal{A},$$ 3.5

and this equation can be studied as a deterministic model of ant colonies displaying excitable dynamics. The model has two equilibrium points, namely ${\mathcal{A}}_0^{\ast }=0$ (no activity spreads) and ${\mathcal{A}}_1^{\ast }=\rho -\alpha /gJ$, associated with persistent propagation. The previous equation is similar to those used in epidemic dynamics [47] associated with a population composed by two classes of individuals (infected and susceptible). Using the density of ants as a control parameter, these two phases are separated by a critical point ρ_c = α/gJ. The global behaviour of this model is summarized in figure 4c where the bifurcation diagram for this system is shown. Above ρ_c an active phase is present, whereas an inactive one is found for ρ < ρ_c.

In this system, the potential function $V(\mathcal{A})$ is

$$V(\mathcal{A})=-\int (gJ\mathcal{A}(\rho -\mathcal{A})-\alpha \mathcal{A})\hspace{0.17em}\text{d}\mathcal{A}$$ 3.6

and is displayed in figure 4d, where we show three examples of its behaviour for different density values. As we already discussed within the context of brain criticality, here too the transition between phases as density is changed involves a shallow potential function, indicating that wide fluctuations should be expected to occur. One remarkable observation from Leptothorax colonies is that they seem to be poised close to the critical density [48] at density levels where theory predicts that maximum information and behavioural diversity is achieved [49,50]. As discussed above within the context of neural tissues, criticality provides a source of fast response and optimal information processing.

The key message provided by this example is that a commonality with other excitable neural systems exists: a universal property is the use of critical points to perform cognitive tasks. Being poised close to critical states provides a natural way of amplifying input signals while remaining most of the time in a low-fluctuation state [51]. Such a compromise makes sense as a way of displaying optimal information while reducing the cost of the system’s state. Is there a well-defined function that can be associated with this? The answer is yes. By using self-synchronized patterns of activity a task may be fulfilled more effectively than with non-synchronized activity, at the same average level of activity per individual [52,53].

(ii). Collective decision-making and symmetry breaking in ant colonies

The next case study involves a classic example of how fluid brains solve a well-defined optimization problem. Specifically, a given ant colony searching on a given spatial landscape needs to discriminate between different available food sources [54–56]. The decision-making rules involved here have inspired major applications within computer science [57–60].

Consider the determination of the shortest path to a single source, to be chosen between two alternatives. This problem can be easily implemented in the laboratory, using a two-bridge set-up (figure 5a). Here the ant nest would be located to the left side and ants would walk through the two-bridge array to reach a food source located on the right side. The two branches can be identical or instead have different lengths. The problem to be solved here is which one is the shortest. Moreover, if one is chosen from a symmetric case (equal paths) how is this choice made. Once again, the solution cannot be found at the individual level: colony-level processes need to be in place to make the right decision.

Figure 5.

Collective decision-making. (a) A two-path experiment allows testing of the mechanisms by which emergent decision-making occurs. The photograph shows an example of a colony that has made a collective decision, as shown by the preferential use of the shortest path. (b) The mathematical analysis of the model associated with this phenomenon shows that two alternative solutions exist associated with the preferential choice of one branch, along with a third one where both branches are used. (c) The parameter space for the simple symmetric case.

This problem can be understood in terms of statistical mechanics where ants are described as Boolean, spin-like variables and the pheromone acts as a field [61]. Here the alternative paths are mathematically mapped to a ferromagnet and the spin-pheromone coupling is described accordingly. In order to introduce noise, explicit use of a temperature is made. In order to present the key ideas under a macroscopic picture, let us here explore a simplified version that can be easily explored analytically.

Ants can use quorum-sensing mechanisms as a way of creating and (responding to) pheromone fields thus generating a large-scale chemical field that allows them to properly make their decision. Initially, ants will walk on both bridges, choosing randomly their branch. We should expect at this point an equal number of ants on each branch, i.e. ρ₁ = ρ₂. However, once an ant has found the food source, it releases a pheromone as it returns to the nest. Other ants will detect the released signal, which helps ants to decide where to move, releasing further pheromones and amplifying the previous mark. The pheromone trail also evaporates, and evaporation will be more effective in the longer trail, where more surface is available. As a result, the shortest path is more likely to be used, and is eventually chosen. Ants have computed the shortest path. A model describing this experiment can be defined as follows. If ρ₁ and ρ₂ indicate the concentrations of trail pheromone in each branch, their dynamics [62] is given by a pair of equations for the pheromone fields:

$$\frac{\text{d}{\rho }_k}{\text{d}t}=\mu q_k{P}_k({\rho }_1,{\rho }_2)-\nu {\rho }_k,$$ 3.7

with k = 1, 2. Here μ is the rate of ants entering each branch, q_i the rate of pheromone production at the ith branch and ν is the rate of evaporation. The functions P_i(ρ₁, ρ₂) can now be understood as probabilities of choosing a bridge depending on the pheromone concentrations. These probabilities are well described by a nonlinear, threshold response function [63,64]:

$$P_i({\rho }_1,{\rho }_2)=\frac{{({\rho }_i+K)}^2}{\mathit{\Theta }\hspace{0.17em}({\rho }_1,{\rho }_2)},$$ 3.8

where $\mathit{\Theta }\hspace{0.17em}({\rho }_1,{\rho }_2)=\sum _{\hspace{0.17em}j=1,2}{({\rho }_{\hspace{0.17em}j}+K)}^2$ and i = 1, 2. The parameter K gives the likelihood of choosing a path free of pheromones (ρ_i = 0).

This is a general model that incorporates attributes associated with each branch. But an interesting scenario arises when one considers the symmetric case where q₁ = q₂ = q. For this situation, the previous set of equations reduces to

$$\frac{\text{d}{\rho }_i}{\text{d}t}=\mu q\frac{{({\rho }_i+K)}^2}{\mathit{\Theta }\hspace{0.17em}({\rho }_1,{\rho }_2)}-\nu {\rho }_i.$$ 3.9

Here there is no true optimal choice: both branches are equal. Now, although the obvious expectation is a similar distribution of ants in each branch, this is not what is observed. We would easily conclude that ants would choose both paths and that individuals will equally walk in both branches. However, what is typically seen is that the symmetry is broken in favour of one of the two branches. Why is this the case? This phenomenon illustrates a very important class of phase transition: the so called symmetry breaking process. Despite the symmetry of the system, amplification of initial fluctuations leads to the formation of a dominant pheromone trail that is used by all ants once established.

The fixed points associated with this system are obtained from dρ_i/dt = 0. One possible solution to this system is the symmetric state ${\rho }_1^{\ast }={\rho }_2^{\ast }={\rho }^{\ast }$ (associated with equal use of both branches), ants equally distributed in both branches. For this special case, we have $P_i({\rho }_1^{\ast },{\rho }_2^{\ast })=P({\rho }^{\ast })=\mu q/2$ and thus we only need to solve a single equation dρ*/dt = μq/2 − νρ*, which gives a fixed point ρ* = μq/(2ν). This is the symmetric state to be broken. The second scenario corresponds to the choice of one of the branches (${\rho }_1^{\ast }\ne {\rho }_2^{\ast }$). Since ρ₁ + ρ₂ = 2ρ* = μq/ν, we see that

$$(\frac{\mu q}{\nu }-{\rho }_i^{\ast }){({\rho }_i^{\ast }+K)}^2={\rho }_i^{\ast }{(\frac{\mu q}{\nu }-{\rho }_i^{\ast }+K)}^2,$$ 3.10

after some algebra, this gives the new fixed points ${\rho }_{+}^{\ast }=({\rho }_{1+}^{\ast },{\rho }_{2-}^{\ast })$ and ${\rho }_{-}^{\ast }=({\rho }_{1-}^{\ast },{\rho }_{2+}^{\ast })$ with

$${\rho }_{i\pm }^{\ast }=\frac{\mu q}{2\nu }+{[{\left(\frac{\mu q}{2\nu }\right)}^2\pm K^2]}^{1/2}.$$ 3.11

This pair of fixed points will exist provided that μq/2ν > K, which allows the derivation of a critical line (figure 5c)

$${\mu }_c=\frac{2K\nu }{q},$$ 3.12

indicating that there is a minimal rate of ants entering the bridges required to observe the symmetry breaking phenomena. For μ > μ_c, the symmetric state becomes unstable (figure 5b,c) while the two other solutions can be equally likely. Below this value, the only fixed point is the symmetric case with identical flows of ants in each branch. This symmetric model can be generalized to (more interesting) asymmetric scenarios where the two potential choices are different (see [55] and references therein) either because the food sources are of different sizes or because the paths are of different lengths and the shortest path needs to be chosen. This symmetry breaking phenomenon has also been observed in the ant colony panic responses [65], army ant trails [54] and optimal group formation [66]. A specially interesting proposal concerning the phenomenon of symmetry breaking in ants was made in Bonabeau [67], where it was suggested that flexible behaviour leading to efficient decisions is more likely to occur close to critical points.

(iii). Task allocation in ant colonies as a parallel distributed process

In the previous example, we considered a set of agents described as binary variables, thus ignoring the combinatorial complexity that should be expected from an insect equipped with a brain. Moreover, it is clear that the active/inactive dichotomy hides a repertoire of potential activities that can be carried out by individuals, associated with the set of tasks needed to maintain the colony. DOL is in fact one of the most important and widespread phenomenon in nature, and very common in social groups [68]. It has been shown that the dynamics of subsets of individuals performing specific tasks within colonies is an emergent phenomenon [10]. In this scenario, a colony that needs to perform a given set of tasks under given environmental conditions (and respond to changes in flexible ways) must be capable of sensing its internal state using some kind of distributed information processing.

Inspired by the dynamics of harvester ants, Gordon et al. [69] proposed a neural network model of task allocation where individual ants are represented by a sequence of Boolean variables instead of a single ON–OFF description. Observations from extensive field work on harvester ants (Pogonomyrmex) show that members of an ant colony perform a variety tasks outside the nest, such as foraging and nest maintenance work. Remarkably, this is a monomorphic species, i.e. individuals exhibit identical phenotypes. The number of ants actively performing each task changes over time due to task switching as well as the presence of inactive workers [70]. As discussed in Gordon [11], interactions among ants involve physical contact. This allows sensing the state of other nest-mates to create a network of information exchanges. Experimental perturbation of the number of ants performing a given task triggers changes in the numbers of individuals performing other tasks. Importantly, this switching dynamics is a consequence of the microscopic, local ant–ant interactions. The attractors associated with normal and perturbed conditions are thus a collective-level outcome of individual interactions.

In their model, Gordon and co-workers consider a set of four observed tasks, namely: patrollers, foragers, nest maintenance and midden workers, displayed by harvester ants. Additionally, individuals can become inactive (as reported in ant colonies, see previous section). Since each type of ant performing any of the four tasks can become inactive, the model assumes that eight possible vectors can represent the available space state which can be covered by an internal state of three binary variables [69]. Specifically, ants are described now as 3-spin vectors $S_k=(S_k^1,S_k^2,S_k^3)$. In their original paper, they use the notation P = active patroller, F = active forager, N = active nest maintenance worker and M = active midden worker. The lower case versions (p, f, n, m) would indicate inactive versions of the previous vectors. The space of possible internal states is indicated in figure 6a,b. These are represented as vertices of a Boolean cube, where all states are, respectively, indicated as strings of + 1 and −1 values.

Figure 6.

Neural network model of task allocation in ant colonies. The dynamics of harvester ants in Gordon et al. [69] can be described in terms of virtual ants (a) each carrying a 3-spin internal description, with changes taking place by means of direct pairwise interactions. The total state space is a three-dimensional Boolean cube (b) where we indicate active (observable) tasks in the top of the cube while a lower layer of inactive states is formed by a flip in the first spin (negative for inactive ants). The model exhibits an attractor dynamics with an associated potential (energy) function. (c,d) The potential function is easily found for a two-task system for the specific values of parameters α = 1 and β = 0.1 (c) and β = 0.5 (d).

The simplest approach for this problem is to assume that the different components of the internal state act independently, with different associated weight matrices. In this way, we would have

$$S_{\hspace{0.17em}j}^{\mu }(t+1)=\mathit{\Theta }\hspace{0.17em}(h_{\hspace{0.17em}j}^{\mu }(t))=\mathit{\Theta }\hspace{0.17em}(\sum _k{J}_{\hspace{0.17em}jk}^{\mu }{S}_k^{\mu }(t)).$$ 3.13

The (internal) state of $S_{\hspace{0.17em}j}^{\mu }$ will remain stable after one interaction, provided that $S_{\hspace{0.17em}j}^{\mu }{h}_{\hspace{0.17em}j}^{\mu }>0$. An energy function is defined accordingly as follows:

$$\mathcal{H}(\{S_k^{\mu },J_{ij}^{\mu }\})=-\frac{1}{2}\sum _{\mu }\sum _{i,j}{J}_{ij}^{\mu }{S}_i^{\mu }{S}_{\hspace{0.17em}j}^{\mu }.$$ 3.14

In the macroscopic realm, the observable state is the number of ants performing each task from the repertoire. It is then desirable to have a description where the energy minimization is defined in terms of the set {n_k}. Thus, the energy function now reads as

$$\mathcal{H}(\{n_k,{\mathit{\Gamma }}_{ij}\})=-\frac{1}{2}\sum _{i,j}{\mathit{\Gamma }}_{ij}{n}_i{n}_{\hspace{0.17em}j},$$ 3.15

with a new set of parameters {Γ_ij} that depend on the microscopic couplings and can be derived from the initial matrices [69]. This energy function allows a description of the system’s equilibrium states (attractors) as a high-dimensional surface the minima of which correspond to the task allocation solutions. This form is consistent with a reaction-based dynamics where pairwise interactions among classes of individuals conditions the global dynamics. As a simple illustration of this idea, let us consider a two-state/two-task case, where it is not difficult to show that the energy function will correspond to

$$\mathcal{H}(N_1,N_2)=-\frac{1}{2}(\sum _{S_i=+1}{S}_i{h}_i+\sum _{S_{\hspace{0.17em}j}=-1}{S}_{\hspace{0.17em}j}{h}_{\hspace{0.17em}j})$$ 3.16

$$=-\frac{1}{2}(\alpha N_1^2+\alpha N_2^2-2\beta N_1{N}_2),$$ 3.17

where we use Γ₁₁ = Γ₂₂ = α and Γ₁₂ = Γ₂₁ = β. We can easily recognize in this solution the elliptic paraboloid, displaying a single minimum. In figure 6c, we show an almost symmetric energy surface for α = 1, β = 0.1, whereas a less symmetric case is displayed in figure 6d, where β = 0.5. In the latter, the coupling between the two tasks creates an elongated valley that would allow for more population fluctuations.

(iv). Collective dynamics of communicating populations

Insect colonies use different organic molecules (pheromones) to transmit signals and process information at a colony level. It is safe to assume that evolution has imprinted on ants a hard-wired pheromone-based detection physiology that generates an internal image of the local environment for each individual ant; however, such a picture is incomplete when confronted with the full complexity of the colony. It is the cobweb of diffusing pheromone signals and ants acting as rewiring agents that confers on the colony its true evolutionary potency. Individual ants are relegated to acting merely as cogs within the macroscopical system [38]. This multiple-scale interrelation is the object of study of the present model by Mikhailov [71].

Imagine a colony of ants individually labelled as i = 1, …, N. Now, introduce two-state variables for each ant as S_i ∈ {−1, +1}, $\mathrm{\forall }i$. Thus, vector S = (S₁, …, S_N) characterizes the full configuration of the system. In this model, ants are again acting as neural agents, but they are also able to send out and receive messages into and from the colony. A message is encoded in a pheromone cocktail, and ants continuously secrete it. To simplify the system’s dynamics, we will consider that a message is fully described with two labels, namely μ_σ,j = (σ, j), where σ ∈ {−1, 1} and j corresponds to the address tag. In other words, message μ_σ,j delivers information σ to the jth ant.

$$S_i(t+\tau )=\text{sign}\hspace{0.17em}(m(+,i)-m(-,i)).$$ 3.18

Let us introduce a correspondence matrix, {ω_ij}, with each of the N(N − 1) elements of the former taking values {−, +}. The function of this matrix is to determine whether a signal will be sent or not in a time interval τ. The way it works is depicted in figure 7c. If ω_ji = +, the sender ant, i, will send a message μ_±,j to ant j only if S_i = ±, whereas for ω_ji = −, the message will be anticorrelated with the state of i, i.e. a message μ_±,j is sent only if $S_i=\mp$. In simpler terms, the correspondence matrix distinguished two channels of information transfer: correlated (ω = +) or anticorrelated (ω = −) message and sender state. On the other hand, we define a frequency distribution, I_ij, as the number of messages per unit time τ that ant j is sending to ant i (figure 7b). Within a spatial context, it is clear that I_ij = I(|i − j|), where $|\cdot |$ is the distance between two ants.

Figure 7.

Collective communication dynamics in ant colonies. In (a), we display an agent i and a set of messages reaching it within time τ, all addressed to i while some carrying the + order others the − order. These messages will be integrated according to equation (3.18). On the other hand, (b) shows how interactions via message sending depends on the frequency (or intensity) of messaging between agents, I. Notice that I values decay with distance. Finally, the way that orders are sent by senders (c) depends on yet another set of couplings {ω_ij ∈ {−, +}}, which determine whether a + or a − order will be dumped into the system depending on the actual state of the sender S_i = ±. Schematically, the arrow connecting sender and receptor is blocked (crossed out) for anticorrelated correlation between coupling ω_ji and sender state S_i.

Let us then consider the dynamics of the messages present in the system with labels (σ, i),

$$\frac{\text{d}m(\sigma ,i)}{\text{d}t}=-\delta m(\sigma ,i)+\frac{1}{2}\sum _{\hspace{0.17em}j}{I}_{ij}(1+\sigma {\omega }_{ij}{S}_{\hspace{0.17em}j}),$$ 3.19

where we have dubbed δ the message decay rate. Therefore, under the stationary regime, we expect

$$m(\sigma ,i)=\frac{1}{2\delta }\sum _{\hspace{0.17em}j}{I}_{ij}(1+\sigma {\omega }_{ij}{S}_{\hspace{0.17em}j}),$$ 3.20

which, combined with (3.18), leads to

$$S_i(t+\tau )=\text{sign}\left(\frac{1}{\delta }\sum _{\hspace{0.17em}j}{I}_{ij}{\omega }_{ij}{S}_{\hspace{0.17em}j}\right).$$ 3.21

Notice that (3.21) is equivalent to the Hopfield model (2.1), provided that ω_ij = ω_ji. Thus, patterns can be stored in a similar manner by following a Hebbian approach by associating:

$$\frac{1}{\delta }{I}_{ij}{\omega }_{ij}\to J_{ij}=\frac{1}{N}\sum _{\mu =0}^{\hspace{0.17em}p}{\xi }_i^{\mu }{\xi }_{\hspace{0.17em}j}^{\mu },$$ 3.22

where, as in §2a, $\{{\xi }_i^{\mu }\}$ will correspond to the agent states of μ = 1, …, p different stored patterns. Although limitations to capacity will also apply here, perhaps more interestingly, other constraints will arise too, namely:

• (1)Agent-to-agent distance dependence on the signal intensity, I_ij = I(|i − j|), which should take the form of a monotonically decreasing function. Effectively, this leads to a diluted network, i.e. every agent does not connect to every other agent.
• (2)Environmental noise: signal loss due to fluctuations of the information channel. This can be formalized as thermal noise, which has also been discussed in §2a.
• (3)Cost-efficiency effects: the address-message system devised here carries with it a large cost on the senders to produce the necessary chemical repertoire so that the signal is well-transmitted with minimal error.

The previous models, with all the simplifications they contain, provide a range of examples of how to approach collective behavioural patterns in ant colonies. The first two examples include amplification processes with two rather different outcomes. In one case, time-dependent fluctuations are observed and are connected to high information transfer. Additionally, the suggested criticality provides a source for low energy use together with a rapid response to perturbations. The second deals with solving a short-path (optimization) task which requires exploiting the well-known phenomenon of symmetry-breaking in order to redirect the population towards the shortest branch. In both cases, the state of the system is described by population values where individual ants have no other identity than either being active/inactive or located in one of the two possible paths.

The two other examples provide an increasingly more defined state for ants. When dealing with a number of tasks that require a distribution of individuals performing them, ants are represented by a Boolean vector and interactions occur following a threshold function (as in standard neural nets) but with a state-dependent choice of links. In other words, in contrast with neurons in solid brains, there is no predefined weight matrix but a state-dependent one. This is in fact an especially important difference, along with the fact that the attractors are given by populations of Boolean strings and any ant can end up in one of them. The fourth model makes a strong attempt to get closer to Hopfield’s picture, but is less realistic in terms of modelling real case scenarios. In order to achieve that result, the model requires incorporating chemical mediators along with the correspondence matrix.

An important message from the previous examples is that both liquid and solid brains share common descriptions (at some level) such as the presence of threshold-like phenomena. They also achieve attractor states but they are smaller in number, and more degenerate in the liquid state that the solid one, even if full connectivity is considered. In the Hopfield model, for example, a very large number of attractors are present, but each neuron will have a specific set of connections. In the next section, we explore the second class of liquid systems as defined by the dynamics of cells in the IS. In this case, the functional constraints are associated with detection and response to information involving signal cascades mediated by interacting sets of cells.

(b). The immune system as a liquid brain

The IS consists of a myriad of chemical compounds (e.g. antibodies, cytokines) and multiple cell lines (B cells and T cells or lymphocytes, macrophages, etc.) aggregated into a multi-component complex system. The essential purpose of the IS is to detect external and malicious agents (antigens) such as viruses, bacteria or cancerous cells, and prompt an appropriate reaction (antigen neutralization or tolerance). At the same time, it must be able to distinguish the latter from internal signals (the self). As such, the IS must be capable of processing, storing and manipulating large amounts of information [72].

The map of interactions of the IS can be depicted as an interwoven web of signalling and response functions between all its agents. Unravelling a full picture of the IS is beyond the scope of this work. For the purpose of our discussion, we will focus on the three core elements that significantly shape the IS architecture: T cells, B cells and antibodies (Ab). Lymphocytes have specific enzymes on their membranes that store a molecular compound that has been randomly generated during its maturation process. This compound binds to specific fragments of proteins (epitopes) coming from an antigen (often through an antigen presenting cell, APC), hence prompting an internal cascade of reactions that activate the lymphocyte. The collection of receptors of a given lymphocyte clone-line is dubbed an idiotype.

Upon detection, B cells (aided by helper T cells) will proliferate thus generating copies of the same receptor structure, while secreting large concentrations of its specific antibody. In summary, the clonal expansion theory [73] states that, since the generated clones share their idiotype, successive binding to the antigen will be triggered and an amplification process will lead to an immune response [74].

On the other hand, a more systemic approach to the IS reveals an underlying network of idiotypes that excite or inhibit one another through the same detection/reaction mechanisms as with antigens. This phenomenon is known as an idiotypic cascade: an initial perturbation (antigen) activates a series of idiotypes filling the system with their corresponding antibodies (Ab1), which, in turn, are detected through another set of idiotypes thus prompting a second batch of antibodies (Ab2), and so on and so forth. This observation suggests a network scheme where each node is associated with an idiotype and each link will correspond to an interaction between any two idiotypes (see later figure 9a–c).

Figure 9.

The IS as a liquid brain. (a) An interaction between an APC carrying a fragment of an antigen and presenting to a lymphocyte (L). (b) Upon matching, the lymphocyte will react by secreting antibodies with the corresponding matching code, thus flooding the system with its idiotypic information and prompting an idiotypic cascade. (c) A representation of the subjacent idiotypic network operating across the IS. This network is actually self-organized into two major blocks (e) of heavily influential (darker region) and weakly influential (lighter region) nodes. Such an effect can be computationally studied by looking at the strength distribution (d), P(ω), noting that picking a random node from the right/left (strong/weak) (i/j) ends of the spectrum, and then looking at its corresponding next neighbours strength ($\overline{i}/\hspace{0.17em}\overline{j}\hspace{0.17em}$), they typically fall under the same category, i.e. strong/weak nodes connect to strong/weak nodes. This suggests a network-like mechanism for tackling the self ($\mathcal{S}$)/non-self ($\mathcal{N}\mathcal{S}$) classification problem (ω-axis is depicted in logarithmic scale). Strong nodes are responsible for self-addressed Ab, and vice versa. Part (d) is adapted from Barra & Agliari [76, pp. 15–16].

Idiotypic cascades were first observed and theorized by Jerne [75] and have since spurred a scientific debate between the allopoietic/autopoietic (reductionist/systemic) approaches to the IS [76–78]. While Burnet’s theory provides some mechanisms for how the IS generates its idiotypic repertoire capable of self/non-self discrimination, Jerne’s network approach complements this process and shows how a distributed computation concatenated to clonal theory might give rise to crucial information-processing aspects of the immune response.

In this section, we will study some fundamental aspects of the IS as a liquid brain. We will begin by looking at the size of the IS and how it is constrained by its fundamental function of antigen detection and discrimination. Then we will study how the IS is capable of storing information at a network level, discuss how it makes use of its idiotypic landscape structure to naturally reproduce a reliable self/non-self classification, and briefly comment on the implications of such a systems-view to the IS.

(i). Simple constraints for the probability of detection

Early studies of the IS showed that epitope reactivity for a generic lymphocyte (B cell or T cell) is of the order 10⁻⁵; in other words, the probability that a random epitope binds to the surface of a lymphocyte is given by p ≃ 10⁻⁵ [74]. This begs the question: why wouldn't the IS organize such that p ∼ 1?

In Percus et al. [79], a simple argument was put forward to show that the fact we observe such values of p might be related to the problem of self/non-self recognition, which strongly constrains the way the IS is assembled.

Consider the following definitions: n is the total number of expressed antibody receptors in the IS repertoire, N is the number of foreign epitopes for a given environment and N^′ denotes the number of self epitopes, or epitopes derived from cells belonging to the organism. Thus, the goal of the IS is to properly distinguish the foreign epitopes while avoiding an immune response to the self-originated ones. Let us denote by P(N, N^′; n) the probability that the repertoire of size n is able to properly detect N foreign epitopes and not detect N^′ self epitopes. Note that the probability of non-recognition of a random epitope for a single lymphocyte is given by q = 1 − p. Hence,

$$\{\begin{array}{ll}{q}^n;& \mathrm{probability}\hspace{0.17em}\mathrm{of}\hspace{0.17em}\mathit{n}\hspace{0.17em}\mathrm{consecutive}\hspace{0.17em}\mathrm{non}\text{-}\mathrm{recognitions}\\ 1-q^n;& \text{probability of at least one recognition.}\end{array}$$

Therefore, we may now compute

$$P(N,N^{\mathrm{\prime }};n)={(1-q^n)}^N{(q^n)}^{N^{\mathrm{\prime }}}.$$ 3.23

The goal is to maximize equation (3.23). This is easily done by maximizing the log P(N, N^′; n), which leads to an optimal value for q,

$$q={(1+\frac{N}{N^{\mathrm{\prime }}})}^{-1/n}\approx 1-\frac{1}{n}\log (1+\frac{N}{N^{\mathrm{\prime }}}),$$ 3.24

where we expanded the previous expression using 1/n ≪ 1. Notice that we can now write

$$p\approx \frac{1}{n}\log (1+\frac{N}{N^{\mathrm{\prime }}}).$$ 3.25

Inman (see [74, pp. 1226–1227] and references therein) estimated the size of foreign epitopes to be of the order N ∼ 10¹⁶, while, for the human genome, if we approximate the number of self epitopes per protein to be of about 10, then N^′ ∼ 10⁶. The IS repertoire (total number of idiotypes) can also be approximated by n ∼ 10⁷ [80], which, by equation (3.25), yields a prediction of p ∼ 2 × 10⁻⁶. This is smaller than empirically obtained values of p, which are of the order p_exp ∼ 10⁻⁵. This might imply that the IS is operating at a non-optimal stage, reacting more often than necessary. This suggests that further mechanisms must be at play in optimizing the immune response. Notice that, due to the logarithmic nature of (3.25), possible miscalculations of N or N^′ will not entail substantial deviations for the predicted value of the detection probability p. For more details on this approach, see Perelson & Oster [81, pp. 656–657 and references therein] and Percus et al. [79].

(ii). Percolation thresholds in the immune system

After Jerne’s discovery of idiotypic cascades, novel ideas were put forward in trying to understand the organizational principles of the IS as a network. Perelson [77] introduced a simple model of the idiotypic cascading phenomenon. Given a repertoire of n idiotypes (i.e. n different types of antibodies), and assuming that paratopes and epitopes can be thought of as bit-strings of size L (figure 8c), then we will consider that an antibody can detect (bind to) a given string if the number of matched pairs of the ordered paratope–epitope interaction exceeds a threshold value, θ < n. As we will see, this readily imposes strong bounds on the system performance.

Figure 8.

Percolation in immune networks. Idiotypic cascades take place at a network level in the IS. (a) A critical percolation cascading on a Bethe lattice of degree z = 3. Concentric circles delimit successive layers of the cascade. (b) The percolation probability depends on the matching threshold θ. At low threshold values, the system is highly connected, allowing deep penetration across layers, while for high θ, the matching probability decays abruptly, leading to a phase of low connectivity with small-sized cascades. Right in the interface, we have the percolation point. (c) Two strings (eptiope-paratope) of length L = 10 with seven matching pairs and three non-matching pairs. For example, if threshold θ = 5, this particular pair of strings would react, whereas for high fidelity matching (θ = 8), the pair would not connect.

Recall that, under Jerne’s paradigm, antibodies are now capable of matching with other antibody types and concatenate into an idiotypic cascade. Thus, we can infer that, for a high threshold value (low reactivity), fewer antibodies will be matching, but also fewer antibodies will be able to detect and react to a given antigen. On the other hand, the reverse is also true: for low values of θ (high reactivity), antibodies will be triggered altogether, as the matching probability is expected to increase. Therefore, it is interesting to study what type of structure will emerge from this simplified model.

Suppose that a given antibody is physically connected to a number of antibodies z, i.e. it will encounter up to z other antibody types but might or might not bind to them. Now, the probability that any pair of antibodies do match is denoted by p, which, by definition, will depend on θ (see below). Thus, given an initial perturbation into the system (such as antigen exposure) then an idiotypic cascade is triggered, where idiotypes react to each other. Such a process will look like a Bethe lattice of degree z (figure 8a). Denote by $\mathcal{A}(i)$ the number of activated antibodies at the ith layer of the tree, then it is easy to show that:

$$\mathcal{A}(i+1)=p(z-1)\mathcal{A}(i),$$ 3.26

which implies that there will be a characteristic probability value p = p_c = (z − 1)⁻¹, at which the network becomes connected, exhibiting a percolation phase transition [36]. For values of p > p_c, the network is fully connected, while for p < p_c, any initial perturbation will eventually die out (figure 8a,b).

For the IS one can argue that z ∼ n in other words, the system is sufficiently fluid and the coarse number of elements is sufficiently large so that any physical interaction can occur. This sets a value on the critical threshold at p_c ∼ n⁻¹. On the other hand, one can compute p = p(θ) by assuming that each bit, of the L-sized strings, is generated by a coin toss. Then, the probability of having two strings with sufficient complementary bit-to-bit values is

$$p=\sum _{k=\theta }^L(\genfrac{}{}{0ex}{}{L}{k}){\left(\frac{1}{2}\right)}^k{\left(\frac{1}{2}\right)}^{L-k}=\frac{1}{2^L}\sum _{k=\theta }^L(\genfrac{}{}{0ex}{}{L}{k}),$$ 3.27

which is plotted in figure 8b. We observe a sudden transition from low to high reactivity at around θ ∼ L/2. In fact, as L → ∞, then p(θ) → 1 − Θ (L/2).

Both n and p have been independently measured ([77, pp. 19–20] and references therein). The repertoire size is estimated to be of the order n ∼ 10⁶, while p ∼ 10⁻⁵. Hence, the IS operates in the post-critical regime, where connectivity is high and large cascading events are common.

(iii). Information storage in immune networks

In the search for a clear understanding of how ISs store and process information, optimization arguments as above do not suffice under the light of Jerne’s theory of idiotypic networks. Initial attempts to describe how information is distributed over the network connecting different idiotypes were put forward by De Boer, Hogeweg, Weisbuch and Perelson (see [74, pp. 1229–1258], and references therein). Here, we will briefly summarize a minimal model by Parisi [78] that involves Hopfield-like NN and imposes global limits on the pattern recognition processes that a distributed network of idiotypes must follow.

Consider the set of antibody binary concentrations {c_i(t) ∈ {0, 1}}, for i = 1, …, N, with N the total number of antibodies of a healthy human IS (around 10⁶ − 10⁷). To all effects and purposes, ‘antibodies’ and ‘idiotypes’ are interchangable from here onwards. Next, we model idiotypic interaction networks, by imposing a dynamical process of idiotype concentrations in the same spirit as (2.1):

$$c_i(t+\tau )=\mathit{\Theta }(\sum _{\hspace{0.17em}j}{J}_{ij}{c}_{\hspace{0.17em}j}(t)).$$ 3.28

Now, the interactions between different idiotypes are mediated by {J_ij}, for which we consider the following properties:

• (i)J_ij = 0, i.e. no idiotype self-interaction is allowed, which is the case for paratope–epitope complementarity matching;
• (ii)J_ij = J_ji, which is a simplification of the Onsager affinity relations between idiotypes,³ log|J_ij| = log|J_ji|;
• (iii)J_ij = U(−1, +1), $\mathrm{\forall }i\ne j$.

Condition (iii) states that the values of the off-diagonal elements of J_ij are taken from the uniform distribution between [−1, 1]. These approximations allow for a derivation of overall limits of distributed storage of information. The system is now described as a spin glass [19,20,82].

Stable solutions for this particular problem turn out to be fully characterized by an average number of pre-assigned concentrations, M. In other words, a generic initial configuration of concentrations will inevitably flow into a stable state by switching concentration values on and off until a pre-assigned configuration of concentration levels is reached. These global stable states act as memory basins similarly to how memory is stored in the aforementioned NN models. Naturally, M < N, thus, we can define α < 1 such that M = αN.

Spin glass theory [82] predicts that, for N ≫ 1, out of the total 2^N possible binary states of the system, and for conditions (i)–(iii), a total of 2^λN patterns can be stored, with λ ∼ 0.3. Withal, we can now try to understand the relation between λ and parameter α.

Let us consider the probability (p_m) of randomly choosing a ‘memorized state’ out of all the possible configurations or, simply, p_m = 2^λN/2^N = 2^−(1−λ)N. However, because only M preassigned concentrations are required to fully describe an attractor, we then expect a number of compatible solutions per stable state. Thus, let us compute the average number of compatible solutions per attractor as

$$\begin{array}{rl}\left(\begin{array}{c}\text{av. no.}\\ \text{of solutions}\end{array}\right)& =p_m\times \left(\begin{array}{c}\text{degeneracy}\\ \text{of config.}\end{array}\right)\end{array}$$ 3.29

$$=2^{-(1-\lambda )N}\times 2^{N-M}$$ 3.30

$$=2^{(\lambda -\alpha )M}.$$ 3.31

Notice that the average number of solutions will be greater or equal to one iff α < λ ∼ 0.3. Essentially, this imposes a bound in M. In other words, if we denote α_c = λ, then for M > α_cN, no equilibrium states are found. Thus, M_c ≡ α_cN is the maximum number of pre-assigned antibody concentrations such that the dynamics imposed by (3.28) flow into well-defined stored patterns. This effectively constrains the memory content that an idiotypic interaction web is able to store.

Parisi’s contributions fuelled the statistical physics and spin-glass approach to the analysis of the IS (see sections below), but also suggested how complex interactions among a large number of elements yield strong constraints in the feasability of IS information processing. This has major implications in that it shows how selective preassures pushing for a reliable IS must go beyond the purely genetic component involved in epitope/idiotope generation. Under this higher-level picture, the capacity for learning and reliably retrieving information from the immune network is limited by fundamental statistical attributes. In the next sections, we will study further the structure of these idiotypic networks and explore how other computational properties can emerge from such a systems analysis.

(iv). Idiotypic networks as liquid neural nets

In what remains of this section, we will outline a model by Barra & Agliari (BA) [76] based on statistical physics of a well-mixed/liquid neural web representing Jerne’s idiotypic network. Let us assume:

• (i)A given clone idiotype is fully characterized by a string of L bits. All idiotypes are of the same size.
• (ii)Each string is obtained from successive, independent coin-tosses with values {0, 1}.
• (iii)The number of cells of a clone-type is sufficiently large so that potential idiotypic interactions are always carried out with their respective intensity values.

Assumptions (i)–(ii) are sensible first approximations to the biological processes the IS undergoes during maturation [72]. On the other hand, a sufficiently high number of lymphocytes per idiotype is not realistic under the light of clonal expansion theory. However, the goal of the BA model is to figure out the overall implications of having an idiotypic network description.⁴

Let us construct an idiotope space ${\mathrm{{\rm Y}}}_L\equiv \{\xi \in {\{0,1\}}^L\}$ spanning all possible strings with bit-size L. Indexes i, j, … ∈ {1, …, N}, with N corresponding to the total number of different clone types in the IS. A priori, a complete repertoire would seem to scale as N ∼ 2^L, however, as we will see, the network constraints will give rise to another scaling behaviour between the repertoire size and epitope/paratope length.

Next, we construct the network following a simple model of chemical complementarity. As usual, let us define a complementarity function:

$$\begin{array}{rl}{\kappa }_{ij}& :=\frac{1}{L}\sum _{\mu =1}^L[{\xi }_{\mu }^i(1-{\xi }_{\mu }^{\hspace{0.17em}j})+{\xi }_{\mu }^{\hspace{0.17em}j}(1-{\xi }_{\mu }^i)]\\ & =\frac{1}{L}(|{\xi }^i\hspace{0.17em}|+|{\xi }^{\hspace{0.17em}j}|-2{\xi }^i\cdot {\xi }^{\hspace{0.17em}j}),\end{array}$$ 3.32

with $|A|\equiv \sum _{\mu =1}^L{A}_{\mu }$. This accounts for the total number of complementary inputs between idiotypes i and j. For example, suppose L = 5, then for ξⁱ = (10101) and ξ^j = (01011), ${\kappa }_{ij}=\frac{4}{5}$ (figure 9b). In turn, this allows the construction of a chemical affinity function

$$\begin{array}{rl}\hspace{0.17em}{f}_{\beta ,L}:\hspace{0.17em}\mathrm{{\rm Y}}\times \mathrm{{\rm Y}}& ⟶\mathbb{R}\\ ({\xi }^i,{\xi }^{\hspace{0.17em}j})& \hspace{0.17em}⟼\hspace{0.17em}{f}_{\beta ,L}({\xi }^i,{\xi }^{\hspace{0.17em}j})=\beta {\kappa }_{ij}-(1-{\kappa }_{ij}),\end{array}$$ 3.33

which is defined as a balance between repulsion and attraction effects of anti-complementary and complementary bit-pairs, moduled by trade-off parameter β ≥ 0. Thus, it will be bounded as −1 ≤ f_β,L ≤ +β, distinguishing two interactive regimes for each pair of idiotypes:

$$\begin{array}{r}\hspace{0.17em}{f}_{\beta ,L}=\{\begin{array}{ll}<0& \text{repulsive regime}\\ >0& \text{attractive regime.}\end{array}\end{array}$$ 3.34

Following these precepts, let us outline how the unweighted network of idiotype-idiotype interactions will unfold. The IS can be arguably approximated as a well-mixed system. This means that, following (iii), any possible physical interaction (B cell/T cell or APC/T cell, etc.) occurs at a sufficiently high rate so that we need only to account for their internal affinity structure. Let us then define p_β,L as the probability that two generic idiotypes display a matching interaction. Consider the following:

• (1)The idiotype strings, {ξⁱ}, are extracted by a successive L random coin-tosses with equal probability for {0, 1} values, i.e. p₀ = p₁ = 1/2.
• (2)The complementarity κ_ij and affinity f_β,L functions fully regulate the interactions. In particular, we define a link between two generic idiotypes (ξⁱ, ξ^j) iff f_β,L(ξⁱ, ξ^j) > 0, i.e. if the pair lays on the attractive regime.

In general, the probability for any two idiotypes to produce a complementarity value, κ, is $P(\kappa )=(\genfrac{}{}{0ex}{}{L}{\kappa })/2^L$. Now, owing to assumption (2) and (3.33), then

$$p_{\beta ,L}=\mathcal{P}\left(\bigcup _{\hspace{0.17em}{f}_{\beta ,L}>0}\kappa \right)=\sum _{\kappa =\lfloor L/(\beta +1)\rfloor +1}^{L}P(\kappa ).$$ 3.35

Since N is the total number of different idiotypes, the emergent network picture will be described by an Erdös–Renyi (ER) graph with degree distribution

$${\varphi }_{\beta ,L}(k)=(\genfrac{}{}{0ex}{}{N}{k})p_{\beta ,L}^k{(1-p_{\beta ,L})}^{N-k},$$ 3.36

the mean value of which corresponds to 〈k〉 ≈ p_β,L N. ER networks display a percolation point at which the system acquires a giant connected component [36]. Typically, this occurs at 〈k〉 = 1, associated with p_c: = 1/N. Next, we explore what regime we should expect the idiotypic network to be in and how this reflects on the IS’s repertoire capacity.

For finite values of L, the shape of the function p_β,L as a function of β is that of a transfer function. Recall that the trade-off parameter β separates the favourably repulsive regime (β < 1) from the favourably attractive one (β > 1), β = 1 corresponding to the symmetric case. Now, if chains (epitopes/idiotypes) are considered to be large, then an amplification process occurs depending on the favourably repulsive/attractive regimes determined by the value of β. Such amplification is reflected on the switch-like behaviour of the connection probability. On the other hand, since the percolation threshold will be of the order of 1/N, even if the system is repulsively favoured (β < 1), it can still easily become fully connected.

Now, consider the three elements that are now coming together: probability of connection, p_β,L; number of idiotypes (or different clones), N; and the average number of connections per idiotype, 〈k〉. While the probability of connection is purely a result of the internal chemical interactions, 〈k〉 is a defining feature of our network. Yet experimental data shows a value of N ∼ 10¹⁸, and a connectivity between idiotypes in mature ISs of $p_{\beta ,L}\sim 3-5\text{\%}$ [83]. This means that we should expect a densely connected network of around 〈k〉 ∼ 10¹².

Moreover, the fact that p_β,L is so low suggests that the system operates at the repulsive regime. That being, we may now compute the relation between epitope size L and number of idiotypes in the BA model by using (3.35) and β < 1. This results in a scaling relation $N\sim \sqrt{L}{e}^{\gamma L}$, with γ < 1, as opposed to the bit-by-bit repertoire size, which would grow as 2^L [76, pp. 5–13].

Hitherto, we have been able to characterize idiotypic networks using only basic assumptions for chemical affinity, which has led to a dense ER. But what kind of computations is this system able to perform? And how does the IS use its autopoietic features to distinguish the self/non-self? To provide an answer to these questions, we ought to look at a fine-grained version of the idiotypic network and inquire into how its interaction intensities are distributed over the net.

(v). Stewart–Varela–Coutinho theory

In their seminal papers Stewart, Varela and Coutinho [84,85] showed that a network systems approach to the idiotypic webs described by Jerne actually displayed two major interactive blocks: a highly connected (strongly interacting) module and a loosely connected (low interactive) one. Such an observation suggested that each module’s activity could correlate to the self and non-self reactions of the IS. More specifically, the strongly connected module acts as an auto-regulated dynamical subsystem that is continuously activated and auto-inhibited; this would correspond to a tolerance response, thus associated with the self ($\mathcal{S}$) stimuli, in a healthy IS. On the other hand, the low interactive module shows a basal activity in the system, but under the presence of a stimulus it will be activated, thus prompting a neutralization response. The latter module is then associated to the non-self ($\mathcal{N}\mathcal{S}$) stimuli. Hence, through this network structural property the vast repertoire of the IS is capable of sorting out the self/non-self.

Although this two-block structure could appear to be the result of an intricate evolutionary process, by following the BA model, a twofold assembly akin to Stewart, Varela and Coutinho’s system is shown to emerge for free. This would suggest a generative mechanism capable of explaining the underlying self/non-self modular structure independently of adaptive drives. Let us briefly explore how this phenomenon takes place at the weighted network level.

(vi). Weighted idiotypic networks and mirror types

The affinity function f_β,L works as a representation of the chemical reactions that take place on the cell surfaces, then, following a simple extension to the concept of interaction, connection matrix J_ij(β, L) can be defined as

$$J_{ij}(\beta ,L)\sim \mathit{\Theta }\hspace{0.17em}[f_{\beta ,L}({\xi }^i,{\xi }^{\hspace{0.17em}j})]\hspace{0.17em}{\text{e}}^{\hspace{0.17em}{f}_{\beta ,L}({\xi }^i,{\xi }^{\hspace{0.17em}j})}.\hspace{0.17em}$$ 3.37

Notice how we still impose a lower threshold of connectance by setting J_ij = 0 for f_β,L ≤ 0, which keeps the previous network picture, while turning on the matrix values smoothly in the attractive regime. The exact values for all the J_ij will depend on each realization of the stochastically generated network of idiotypes {ξⁱ}, thus, it will be necessary to normalize each interaction parameter over all the space of possible networks [76, pp. 13–20].

Once the interaction intensities are in place, one can look at the total strength for each idiotype (node) as

$${\omega }_i(\beta ,L)=\sum _{\hspace{0.17em}j|J_{ij}>0}{J}_{ij}(\beta ,L).$$ 3.38

Heuristically, J_ij values characterize the robustness of a given i − j interaction, while ω_i measures how influential idiotype i is relative to the whole network. Now, consider the weight frequency distribution P(ω), which can be shown to be well approximated by a normal distribution [76, pp. 13–20]. If we select an idiotype i and look at its first-order neighbours that inhibit i, namely mirror-i idiotypes (or simply $\overline{i}$), then it is possible to study how the system self-classifies these pairs into two major classes (figure 9e): strong and weakly interacting pairs. The fact that the interaction is symmetrically strong/weak for each pair is a consequence of chemical complementarity in the affinity function.

However, this simple realization turns out to be an extremely powerful tool to resolve the self/non-self distinction. In summary:

• —The P(ω) degree distribution separates the two regimes of strong/weak influential nodes (figure 9d). The weak nodes (blank triangles) happen to have weak mirror types (blank squares) whereas the strongly interacting nodes (reversed blank triangles) have mirror types (black squares) that also are highly interactive nodes. This mechanism gives rise to the $\mathcal{S}/\mathcal{N}\mathcal{S}$ modules.
• —The strong block is hypothesized to account for the self-directed antibodies, while the weak module acts as a basal signal only activated by the presence of non-self antibodies (triggered by external antigens).
• —Establishment of robust memories occurs more effectively at the weakly interacting block, as relative variations in the affinity values will produce more durable configuration changes in this network module.
• —Autoimmune diseases can now be understood as deviations from the two-module structure, where strongly interacting circuits (responsible for self-addressed antibodies) may deviate towards lower weighted regions of the spectrum, thus triggering auto-immune response.

Hence, a natural mechanism for fundamental computational questions such as the self/non-self identification is derived from first principles. These are constructed under the assumption that the interaction time scales are small compared with the global observational time scales, while, on the other hand the ability of the ‘neural agents’ (idiotypes) to rapidly propagate throughout the environment ultimately allows the characterization of the idiotypic network as a biologically meaningful system. Thus, the IS appears to be a limit case scenario for ‘liquid brains’, where it is precisely the high levels of agent mobility that give rise to its capacity to solve classification problems.

This realization leads to novel questions: are there size limitations for ISs and their performance in terms of physical embodiment? What are the consequences of these constraints to the self/non-self distinguishability? Or, in general, can different ‘ISs’ exists across multiple scales?

(vii). B cell and T cell interactions as a glassy system

Immune systems seem to conceal within a concoction of networks the nodes (antibodies and lymphocytes) of which can be active or inactive in relation to their presence or concentration levels. New approaches based on lymphocyte–lymphocyte interactions have shed light onto the intrinsic glassy mechanisms behind IS information processing [86,87].

The previous analysis has only focused on the presence–absence of Ab types in an idiotypic context, yet, underlying the dynamics of idiotypic cascades, B cell clone types are behind the production (or lack thereof) of the antibodies. This will entail the global state of the IS and, as discussed above, allow for self/non-self distinguishability or memory storage. On the other hand, T cells, helpers and suppressors, will promote and inhibit the proliferation of B cells. From a network perspective, this can be understood as a core bipartite network with T cells on one layer and B cells on the other. However, this internal network will contain both excitatory and inhibitory interactions. Such a rich dynamical picture is what ultimately confers the IS its glassy behaviour.

Under this framework, more profound questions can be formally pursued. In Agliari et al. [86], the authors showed how a spin-glass approach to the coupling between lymphocytes leads to different scenarios. In these, the overall clonal expansion extent and the ratio between the number of B cells and the number of T cells (helpers and suppressors) strongly define the regions of stable memory storage and inability to retrieve memories. This has major implications for complex diseases such as HIV. Here, T-cell levels are depleted by the virus and the system is driven into the spin-glass states where the IS fails to operate reliably [86].

Finally, this provides a yet deeper analogy between liquid brains and solid neural networks (see Hopfield model in §1). In the latter, the memory capacity α measures the relative number of patterns stored in a network of a given size, but in the IS this seems to be constrained by the ratio in gross numbers between the two type of lymphocytes. Nonetheless, there is yet another fundamental distinction between the two systems. Hopfield nets operate by interpreting a pattern at a time (provided an input, multiple outputs can be reliably retrieved), but the IS computational task works in a different way; it needs to carry out parallel tasks (as the organism is exposed to multiple pathogens at once), but its decision boils down to a single bit of information: immune reaction or tolerance. This imposes severe restrictions on both the idiotypic and the lymphocyte networks of interactions (see [88] for an excellent investigation into this matter).

4. Discussion

The emergence of cognition in our biosphere has been marked by several key events that allowed the evolution of special classes of cell phenotypes along with ways of wiring them together. Nerve cells and nerve nets pervade the revolution towards new life forms capable of dealing with non-genetic information in complex ways. But the basic ingredients for the emergence of complex forms of information processing have appeared multiple times at different scales and in different evolutionary contexts [2]. Neural-like processing systems have evolved as specialized organs but also as communities of moving agents. In both cases, agent–agent interactions involve some sort of recognition, internal communication coding and stimuli thresholds that decide if changes are made. As shown in previous sections, simple models can capture relevant phenomena associated with both classes of systems.

These two classes of networks share emergence as a major feature. Memory, learning or decision-making are grounded in a set of bottom-up phenomena where emergent properties arise from individual, microscopic interactions (figure 10). Collective phenomena and a physics approach becomes a natural common field from where to extract universal features. These emergent traits can be attractor basins associated with memory states or efficient task allocation, but can also be phase transitions due to the presence of critical connectivities, or even criticality itself, enabling rapid and efficient information transfer.

Figure 10.

Multiscale dynamics in liquid brains. As occurs with many other complex systems, each example of liquid brains involves several scales of description. (a) Ant colonies perform diverse functionalities, such as collective foraging (aiii) on a colony-level basis. At a smaller scale, pairwise interactions among ants take place (aii). Such interactions are localized and, thus, constrained by spatio-temporal properties such as agent mobility or density. At the top of this hierarchy (ai), we encounter single ants as a system. These agents will be defined by a set of rules that drive their behaviour at this minimal scale. (b) A similar scheme can be made for the IS. Scales now involve the idiotypic (or antibody type) network (biii), where information is processed, for instance, at the self/non-self discrimination level (see above). As we zoom in, we encounter the cellular-scale interactions level (bii), which are also associated to the simple-matching recognition dynamics. Finally, yet another level of complexity is reached at the description of the IS elementary agents (bi): viruses, paratopes, epitopes and surface receptors.

What is missing from our previous models? An important piece of complexity that has been ignored in this discussion is the internal complexity of the agents. This is not necessarily a limitation. When dealing with complex systems, we purposely ignore unnecessary detailed descriptions of the system in order to render the problem solvable and provide true understanding. The level of simplification is imposed by the kind of question being addressed. As in the Hopfield model and other classical neural network approaches, neuronal complexity is reduced to the minimal description. The IS too is a rather sophisticated system, exhibiting a considerable diversity of cell types and interactions. Ants, on the other hand, are not simple strings of Boolean bits. Individual cognition exists: ants carry actual brains inside their heads, even if small ones [89].

As pointed out by some authors [90] even if these brains are orders of magnitude smaller than ours, they exhibit some types of cognitive skills. How important are they compared to colony-level cognition? This is an open question that will require further attention. It might be the case that increases in the cognitive complexity of ant colonies is accompanied by reductions in an individual’s cognition. Such a trade-off has been explored in other contexts, like the evolution of multicellularity [91], described by the term ‘complexity drain’. Previous work used coupled discrete maps to suggest that collective computation might also display this phenomenon [92]. Further multiscale models of liquid brains should be developed to properly address this question.

What other systems can be described as liquid brains? As mentioned before, computations arising from gene–gene regulatory links within cells have been studied using similar formal schemes, from Boolean tables to threshold networks [93,94]. Several observations also suggest that gene networks might be critical [95–98].

A final comment needs to be made concerning the physical phases used to present our classification between liquid and solid. The use of the term ‘liquid’ to label the classes of systems discussed here is only partially appropriate. Particularly in relation to ants, their collective dynamics is more appropriately described as ‘active matter’: ants (as well as bacterial and robotic swarms) need to be understood as interacting self-propelled robots [99]. Here too the statistical physics approach has played a key role in understanding coordinated behaviour and its transitions. Once again, in spite of considerable differences, deep analogies exist between classical equilibrium statistical physics systems and those made of active units. Understanding the cognitive complexity of liquid brains and its limits can provide deep insights into the evolution of information-processing, computational systems grounded in living structures.

Data accessibility

This article has no additional data.

Authors' contributions

All authors built and analysed the mathematical models. All authors gave final approval for publication.

Competing interests

We have no competing interests.

Funding

This study was supported by the Botin Foundation, by Banco Santander through its Santander Universities Global Division, FIS2015-67616-P and by the Santa Fe Institute. J.P. acknowledges support from ‘María de Maeztú’ fellowship MDM-2014-0370-17-2. This work has also counted on the support of Secretaria d’Universitats i Recerca del Departament d’Economia i Coneixement de la Generalitat de Catalunya.