Integrated Ising model with global inhibition for decision-making

Significance

The main theoretical framework for modeling decision-making processes has been based on the highly successful drift-diffusion model. However, recent observations challenge this model, indicating that neuronal inhibitory tone increases during difficult discrimination tasks. Motivated by this observation, we introduce a theoretical model whereby neurons are represented by interacting spins, including global inhibition. Compared to experimental results, this model suggests that the brain’s decision-making activity is in proximity to a critical transition line between the ordered and disordered phases. Within this critical region, the model predicts that the decision-making process has unique dynamics and advantageous properties.

Abstract

Humans and other organisms make decisions choosing between different options, with the aim of maximizing the reward and minimizing the cost. The main theoretical framework for modeling the decision-making process has been based on the highly successful drift-diffusion model, which is a simple tool for explaining many aspects of this process. However, recent observations challenge this model. It was found that inhibitory tone increases during situations of difficult discrimination tasks, but the origin of this phenomenon is not understood. Motivated by this observation, we extend a recently developed model for directional decision-making of animals moving in real space. We introduce an integrated Ising-type model that includes global inhibition and use it to describe two-choice decision-making. This model can explain how the brain may utilize inhibition to improve its decision-making accuracy. Compared to experimental results, this model suggests that the regime of the brain’s decision-making activity is in proximity to a critical transition line between the ordered and disordered phases. Within the model, this observation can be explained by noting that this critical region has unique dynamics that give rise to advantageous properties for the decision-making process.

Decision making is a dynamic cognitive process that results in the selection of a course of action or formation of a categorical choice (1). The theoretical description of the decision-making process has been attempted on several levels. There are models that describe neuronal networks that include both excitatory and inhibitory neurons and their dynamics (2–5). On a more abstract level, there is the successful drift-diffusion model (DDM), which assumes that the difference in evidence that is accumulated for each of the options drives the decision process (6, 7). Within this model, decision-making is described by a stochastic diffusion process of a “decision variable” (DV), in addition to a drift which represents the net external evidence in favor of one of the options. In each decision-making process, the DV moves according to the drift-diffusion dynamics until it reaches one of two thresholds, which encode the two abstract alternatives, and a decision occurs (8). The DDM describes the main properties of observed decision-making dynamics and explains the principles of the speed–accuracy trade-off (5, 9). However, due to the simplicity of the DDM, there is only one dynamic parameter that controls many of the results, and it is not able to explain more intricate effects without ad-hoc assumptions and introduction of across-trial variability in the model parameters (6, 10, 11).

Recent observations found an essential role for global inhibition during the decision-making process, with higher levels of the inhibitory neurotransmitter (GABA) detected in more difficult discrimination tasks (12). Considering the role of inhibition motivated us to develop an Integrated Ising model (IIM), inspired by a recently proposed Ising model for animals making directional decisions while moving in space (13–15). In this model, the real-space targets which the animal aims to reach are represented by groups of Ising spins that interact ferromagnetically within each group, but their intergroup interactions become inhibitory for large relative angles. The spins in this model, which are either 1 or 0, represent the firing state of neurons or groups of neurons [as in the Hopfield model (16)]. The model successfully predicted the bifurcations during collective motion of animal groups (13), and single animal movement in space toward static targets (14, 15) or moving conspecifics (17). The Ising model (18) was previously applied to study cognitive processes and the behavior of neural networks during decision-making (11, 16, 19–21) and memory retrieval (16, 22).

Here, we extend the spin-based spatial decision-making model (13, 14) to include the effects of global inhibition as a biologically inspired mechanism and use it to describe the abstract decision-making process in the brain. Our IIM gives us an underlying mechanism that drives the random walk dynamics of DV. However, unlike the DDM, which relies on simple Brownian random-walk diffusion, our Ising model has an ordered phase in which the random walk changes from simple diffusion to run-and-tumble (RnT) dynamics near the transition line (13). Note that this is not a neural circuit model of decision-making that accounts for the dynamics of neural populations in a detailed manner. In line with previous models of decision-making, such as the DDM, it is a biologically inspired effective model of this cognitive process, which could lead to plausible neural circuit implementations in the future (2, 3).

In particular, we find that the regime close to the phase transition within the ordered phase may have advantageous properties for decision-making and can explain the observed role of global inhibition in this process. By comparing our model with different two-choice decision task experiments, we demonstrate that the IIM in the critical regime can explain the observations, such as the relation between error and reaction time (RT) and the effects of increased global inhibition (related to the measured GABA signal).

Theoretical Model

Similar to the regular DDM, we relate the decision-making process to an abstract decision variable (DV) which integrates neuronal firing over time (23, 24) and moves between two fixed and equal thresholds, each encoding one of the options of the two-choice task (Fig. 1A). The major difference between the IIM and DDM is that the dynamics of the DV are not given by drift with diffusion but are calculated from the instantaneous firing states of two neuronal populations that are represented by Ising spins.

Fig. 1.

The IIM for binary decisions. (A, i) The spin system is divided into two equal groups (green and orange), corresponding to the two options of a decision task. The darker circles represent spins that correspond to firing neurons ($\sigma =1$, “on”), while the light circles represent spins corresponding to nonfiring neurons ($\sigma =0$, “off”). The spins in the same group have excitatory interactions (${\mathrm{J}}_{\mathrm{in}}>0$), while spins from one group suppress the other group via cross-inhibitory interactions (${\mathrm{J}}_{\mathrm{out}}<0$). The instantaneous velocity in the stochastic decision process depends on the current system’s configuration ($V=(N_1^{\mathrm{on}}-N_2^{\mathrm{on}})/N=n_1^{\mathrm{on}}-n_2^{\mathrm{on}}$, given by Eq. 1) and is integrated to provide the decision variable (DV). (A, ii) Typical trajectory of the DV in the IIM. The decision occurs when the DV reaches one of the thresholds (green and orange horizontal lines) representing the two options, respectively. The initial conditions are $\text{DV}=0$, and all spins are in their off state. (B) Phase diagram of the IIM. The blue line denotes the second-order transition (solution of Eq. 4), bounding the ordered phase (which has both ballistic behavior at low $T$, $\eta$, and run-and-tumble behavior close to the transition line). The red line denotes the first-order transition, bounding the intermittent phase. Outside these regions, the system is in the disordered phase (equivalent to the DDM behavior). The black circle denotes the tricritical point: ${\eta }_{\text{tri}}=0.439$, $T_{\text{tri}}=0.333$. The green stars denote the parameters for which the velocity distribution and samples of dynamics are shown in panels C–F. (C–F, i) The velocity distribution obtained from multiple decision trajectories. (ii and iii) An example of the time evolution of the DV’s velocity and DV during one decision process (no preference, ${ϵ}_{1,2}=0$). In the ordered phase: $T=0.3$, $\eta =0.05$; the intermittent phase: $T=0.15$, $\eta =0.45$; near the tricritical point: $T=0.29$, $\eta =0.45$; in the disordered phase: $T=0.4$, $\eta =0.6$. The dark green and brown lines in (i) and (ii) indicate the positive and negative MF velocities, respectively (solutions of Eq. 4), while the purple line is the average velocity (which is zero in the absence of any preferences). The horizontal green and orange lines in (iii) indicate the decision thresholds.

We assume that the decision-making circuit can be described in an abstract manner by a network of $N$ spins, divided into two equal competing groups of spins which encode the two options (Fig. 1A, i). Each spin represents a single neuron (or a group of neurons) in the brain, which can be in either one of two states: on or off, ${\sigma }_i=1,0$, corresponding to neurons in the firing or resting state, respectively, such that $N_{1,2}^{\mathrm{on}}$ is the number of on-spins in each group. We assume that interactions between the spins are excitatory within the same group and inhibitory between the two groups. The network is fully connected, and we neglect the spatial organization (21).

In the IIM, the instantaneous velocity with which the DV evolves is given by the difference between the fractions of on spins in the two groups: $V=n_1^{\mathrm{on}}-n_2^{\mathrm{on}}=(N_1^{\mathrm{on}}-N_2^{\mathrm{on}})/N$, where $1,2$ denote the spin group. The position of DV is the time integral over this speed toward either one of the threshold values ($L$, Fig. 1A, ii).

We can relate the spin states to the neuronal firing activity and see how it depends on the global inhibition. Therefore, we can make some predictions using our model with respect to the neuronal activity during decision-making (25, 26). The dynamics of the fractions of on-spins $n_1^{\mathrm{on}}$, $n_2^{\mathrm{on}}$ are described by the following dynamical equations

$$\begin{array}{c}\hfill \left\{\begin{array}{c}d{n}_1^{\mathrm{on}}/dt=r_1^{\mathrm{on}}{n}_1^{\mathrm{off}}-r_1^{\mathrm{off}}{n}_1^{\mathrm{on}}\hfill \\ d{n}_2^{\mathrm{on}}/dt=r_2^{\mathrm{on}}{n}_2^{\mathrm{off}}-r_2^{\mathrm{off}}{n}_2^{\mathrm{on}}\hfill \end{array}\right.,\end{array}$$ [1]

which depend on the rates at which off-spins in each group turn on ($r_{1,2}^{\mathrm{on}}$) and on-spins turn off ($r_{1,2}^{\mathrm{off}}$). These spin-flipping rates can be derived from the effective energy functional that describes the interactions between the spins, representing the excitatory and inhibitory interactions between the neurons. This energy functional includes global inhibition that equally affects all neurons involved in the decision-making circuit, promoting them to revert to their resting state (12), similar to an external magnetic field applied to a system of magnetic spins. In the language of the Ising spin model, this energy functional is written as the following Hamiltonian

$$\begin{array}{c}\hfill \mathcal{H}=-\frac{1}{N}\sum _{i\ne j}{J}_{\mathit{ij}}{\sigma }_i{\sigma }_j+\eta \sum _i{\sigma }_i-{ϵ}_1\sum _{i\in I}{\sigma }_i-{ϵ}_2\sum _{i\in II}{\sigma }_i,\end{array}$$ [2]

where $J_{\mathit{ij}}$ is the coupling constant, such that $J_{\mathit{ij}}=+1$ (ferromagnetic interaction) if spins $i$ and $j$ are in the same group, and $J_{\mathit{ij}}=-1$ (antiferromagnetic interaction) if they are in competing groups (27–29). We assume here that $J_{\mathit{ij}}$ is symmetric for simplicity and show the effects of asymmetric interactions in SI Appendix, section S1). The first term in Eq. 2 is the summation over all distinct pairs of interacting spins. The second term represents the global inhibition field of strength $\eta >0$, which favors the spin state off. The last two terms introduce the strength of preference toward one of the two options (${ϵ}_{1/2}>0$), applied as excitatory external fields. These preferences represent the effect of the evidence that was obtained in favor of one of the options, either in the form of sensory inputs or in the form of retrieved memory gained by a learning process over previous trials.

The spin-flipping rates are taken to be in the Glauber form (30)

$$\begin{array}{c}\hfill \begin{array}{cc}{r}_1^{\mathrm{on}}=\frac{r_0}{1+exp\left(\frac{-2V+\eta -{ϵ}_1}{T}\right)};\hfill & r_1^{\mathrm{off}}=\frac{r_0}{1+exp\left(\frac{2V-\eta +{ϵ}_1}{T}\right)}\hfill \\ r_2^{\mathrm{on}}=\frac{r_0}{1+exp\left(\frac{2V+\eta -{ϵ}_2}{T}\right)};\hfill & r_2^{\mathrm{off}}=\frac{r_0}{1+exp\left(\frac{-2V-\eta +{ϵ}_2}{T}\right)},\hfill \end{array}\end{array}$$ [3]

where $T$ is the parameter that describes the ratio between the strength of noise in the system and interactions between the spins, playing the role of temperature in the model (31, 32). The rates in Eq. 3 are multiplied by a constant $r_0$, which defines the time units, and we set it to $1$.

The changes in the spin system’s configuration depend on the two global control parameters of the model: the noise ($T$) and global inhibition ($\eta$). The spin dynamics are described by a stochastic process, where spin flips occur randomly at the rates given by Eq. 3, which we simulate numerically using the Gillespie algorithm (33) (SI Appendix, section S2.B). The velocity distributions, shown in Fig. 1C–F, i, are obtained from running numerous simulations of the decision trajectories (examples are given in Fig. 1C–F, ii and iii). The peaks in these distributions correspond to the mean-field solutions (see below), while their spread is determined by the finite size of the spin groups (13).

While the spin dynamics are stochastic, we can treat the dynamical equations (Eq. 1) in terms of the mean values of the proportions of on-spins and obtain the mean-field (MF) equations that allow us to map analytically the collective spin phases (Fig. 1B). Using the equations (Eq. 1), we obtain the MF equation for the velocity of the DV

$$\begin{array}{c}\hfill V=\frac{1}{4}\frac{sinh\left(\frac{4V+{ϵ}_1-{ϵ}_2}{2T}\right)}{cosh\left(\frac{2V-{ϵ}_2+\eta }{2T}\right)cosh\left(\frac{2V+{ϵ}_1-\eta }{2T}\right)}.\end{array}$$ [4]

Expanding Eq. 4 in the absence of any preference (${ϵ}_{1,2}=0$) up to the third order at $V=0$, we get the condition for the phase transition, at which the zero solution becomes unstable. It gives the second-order transition line on the phase diagram (the blue line in Fig. 1B)

$$\begin{array}{c}\hfill \eta =T\mathrm{arccosh}\left(\frac{1-T}{T}\right).\end{array}$$ [5]

The area under the blue curve (Fig. 1B) refers to the ordered phase, where one of the spin groups prevails while the other group is inhibited. It corresponds to two stable nonzero solutions for $V$ (Fig. 1C, i). The other transition line, of first-order nature (the red line in Fig. 1B), defines a phase (between the red and blue lines) where the zero solution $V=0$ coexists with two nonzero solutions (Fig. 1E, i). We find the red transition line by solving Eq. 4 and $d$(Eq. 4)$/dV=0$ simultaneously. The area above the blue and red transition lines refers to the disordered phase, where only $V=0$ is stable (Fig. 1D, i).

The tricritical point indicates the place where the second-order phase transition curve meets the first-order phase transition curve. We find it by solving simultaneously Eq. 4, $d$(Eq. 4)$/dV=0$, and $d^3$ (Eq. 4)$/d{V}^3=0$, which gives $T_{\text{tri}}\approx 0.333,{\eta }_{\text{tri}}\approx 0.439$ (the black point in Fig. 1B).

Having mapped the phases of the IIM, we now explore the dynamics of the DV in each of these phases, driven by the stochastic dynamics of the spins. Note that, unlike the DDM, where the stochastic noise in the evolution of DV arises from an independent Gaussian source, in the IIM, the stochastic dynamics are driven by the spin flips, which depend on their mutual interactions.

In the low temperature and inhibition regime of the ordered phase (below the blue transition line), the DV moves mostly in a ballistic-like trajectory (Fig. 1C, ii and iii), where one group “wins” and inhibits the other. As the system’s temperature and inhibition approach the critical values, the positive and negative solutions for the velocity converge to the zero solution $V=0$, and the motion transforms into regular diffusion in the disordered phase (Fig. 1D, ii and iii). In the intermittent phase, the velocity has both zero and nonzero solutions (Fig. 1E). Within the ordered phase, but close to the blue transition line, we find that the dynamics of the DV are of the RnT type, as shown, for example, at the tricritical point (Fig. 1F). The stochastic dynamics of the spin flips allow the system to switch between the MF solutions, giving rise to the RnT behavior.

The initial conditions for the simulations shown in this paper were for the spins in their zero configuration. In SI Appendix, section S2.F, we show the results when the initial conditions are such that the spins are in a random initial configuration. The results are not sensitive to this choice of initial conditions unless the system is at very low temperatures.

Comparison with Other Decision-Making Models

We now briefly compare the IIM with a few similar or commonly used models for decision-making (Fig. 2). In the IIM, $n_1$ and $n_2$ indicate the instantaneous firing states of the two spin populations that refer to the two neuronal populations (Fig. 2A). The spins interact via self-excitation within each group (${\mathrm{J}}_{\mathrm{in}}$) and cross-inhibition between the groups (${\mathrm{J}}_{\mathrm{out}}$). The spins tend to fire at higher rates in the presence of evidence in favor of one of the options (${ϵ}_1$, ${ϵ}_2$). The global inhibition affects both groups, promoting the off-state. $y_1$ and $y_2$ represent the integrated quantities, where $y_1=\text{DV}=\int (n_1-n_2)dt$ and $y_2=-\text{DV}$. The decision is made when either $y_1$ or $y_2$ reaches the fixed threshold $L$.

Fig. 2.

Architectures of different decision-making models. (A) IIM model. (B) IDM model (11). (C) LCA model (4). The green arrows indicate excitation. The purple lines with flat ends indicate cross-inhibition. The brown arrows or lines with flat ends indicate global inhibition (IIM), activation threshold (IDM), and leakage (LCA).

Another theoretical model for decision-making that is based on the Ising model in a similar spirit to our work is the Ising decision maker (IDM) (11). In the IDM (Fig. 2B), the neural network consists of two pools of neurons (represented by their instantaneous firing states $n_1$ and $n_2$) with pairwise excitatory (inside the group, $w_{1,2}^{+}$) and inhibitory (between the groups, $w_{12}^{-}$) interactions, and activation threshold $\theta$. The external fields $b_{1,2}$ represent the sensory evidence. The system starts at a random low-activity state. In the ordered phase of the Ising model, there are two minima that correspond to the states, with one of the two spin groups having a large activity while the other group is inhibited. The decision in the IDM is made when the system reaches the region around one of these minima, corresponding to one group reaching a high-activity state (25) (SI Appendix, section S3 and Fig. S6A). In the disordered phase of the Ising model, the IDM becomes locked in indecision (SI Appendix, Fig. S6 B and C), and therefore, does not approach the DDM dynamics in any regime, unlike the IIM. This behavior arises due to the major difference between the IIM and the IDM, where the IDM does not perform integration of the firing activity of the spins over time (Fig. 2 A and B).

Another common class of decision-making models is the leaky competing accumulator (LCA) model (4). The model’s structure is shown in Fig. 2C. In this model, there are two accumulator units ($y_{1,2}$) that integrate the noisy evidence from two firing neuronal populations ($I_{1,2}\pm c)$. The accumulators interact via cross-inhibition ($w$). The model also allows the decay of the accumulator’s activity (“leakage,” $k$). The decision is made once the activity of one of the integrated quantities reaches a positive threshold $L$. The main difference between the LCA and the IIM is that in the LCA, the cross-inhibition appears only at the level of the integrated firing rates, and there is no cross-inhibition at the level of the underlying firing units, as in the IIM. The lack of cross-inhibition at the underlying firing signal that enters the integrator means that in the LCA, there is no sharp phase transition associated with the decision-making process.

The comparison between the IIM and the DDM was mentioned above (see also SI Appendix, section S4.C). We note that in the disordered regime, the IIM recovers the DDM behavior. In this respect, the IIM extends the DDM by introducing correlations in the dynamics arising from the underlying spin interactions. A crucial difference is that in the DDM, the dynamics of the firing that the DV integrates are purely Gaussian white noise (with an additional drift) with no temporal correlations. In contrast, in the IIM, the dynamics have temporal correlations induced by the Ising coupling between the spins. The temporal correlations appear most clearly in the RnT trajectories of the DV for the ordered and intermittent phases (Fig. 1C, E, and F).

Note that in the spin models (IDM and IIM), the noise in the dynamics of the spin populations is driven by the stochastic nature of the spin-flips, determined by the internal noise parameter $T$. The external noise is manifested in the magnitude of the evidence strength (${ϵ}_{1,2}$ in IIM and $b_{1,2}$ in IDM; see Fig. 2), such that lower signal-to-noise ratio corresponds to lower value of the evidence. In contrast, in the DDM, the external noise is explicitly added to the input from the stimuli (drift) in the form of diffusion.

Our model, therefore, contains the phase-transition property of Ising-based models at the neuronal firing level (as in the IDM), while the decision is made at the level of an integrated quantity, similar to the DDM and LCA models. Combining these properties, our model extends previous models and exhibits unique dynamical regimes and decision-making properties.

Properties of Decision-Making Processes in the IIM

We now explore the characteristics of the IIM decision-making processes in the presence of evidence, which we take here to favor only the option represented by the threshold at $DV=+L$ (${ϵ}_1\ge 0,{ϵ}_2=0$). The velocity distribution, obtained from multiple decision trajectories, is shifted by the evidence toward higher speeds in the direction of the preferred option (“correct” decision) Fig. 3A, i. The simulated trajectories of the DV in the presence of preference allow us to extract the probability of arriving at the correct decision and the distribution of “RT,” which is the time until a decision is made when the DV reaches either of the two thresholds at $\pm L$. Throughout the paper, by RT, we mean the average reaction time, and the error rate is the fraction of processes where the DV reached the unfavorable negative threshold at $-L$. Since we want to explore the dependence of the decision-making process on the dynamics of the DV in different parts of the IIM phase diagram, we fix the value of $L$ (see SI Appendix, section S2.E and Figs. S4F and S5 A and B for other values). Note that in the IIM, the RT is given in units of the spin-flip time constant $1/r_0$ (Eq. 3).

Fig. 3.

Decision-making dynamics in the IIM. (A, i) The probability density of velocity in the IIM near the tricritical point ($T=0.29,\eta =0.44$), in the presence of a weak constant preference (blue contour, ${ϵ}_1=0.01$) and without (gray contour, ${ϵ}_1=0$), obtained from multiple decision trajectories. The purple vertical line indicates the average velocity. The green and brown vertical lines indicate the MF solutions of Eq. 4, which coincide with the blue distribution’s peaks. The dashed lines denote the same quantities for the unbiased case. (A, ii) Calculated decrease in the error rate as a function of increasing evidence strength (${ϵ}_1$) at $T=0.3$. As the global inhibition ($\eta$) increases, the error rate decreases. (A, iii) Mean RT as a function of the evidence strength (colors as in A, ii). The RT increases with increasing global inhibition. In the Inset, we plot the RT as a function of the error rate for the two cases of $\eta =0.1,0.4$. (B, i) Error rate, (C, i) RT, and (D, i) the RT ratio in the correct and wrong decisions (${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$) as functions of the system’s parameters ($\eta ,T$) at fixed ${ϵ}_1=0.01$, presented as heatmaps. The red and blue lines denote the first- and second-order transitions, respectively (Fig. 1B). (B, ii) Error rate, (C, ii) RT, and (D, ii) the ratio ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$ at $\eta =0$ as functions of temperature $T$ (and same ${ϵ}_1$ as above). The blue vertical line indicates the critical temperature ($T=0.5$) corresponding to the second-order phase transition. We denote three regimes on the ratio: zone III is above the transition line (the disordered phase). In the ordered phase (zones I, II), we denote a change in the trend of the ratio ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$ by the vertical black line. In zone I, the ratio ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$ decreases with increasing $T$, while in zone II, it has a nonmonotonous dependence on the temperature. The black curve in (C, ii) and (D, ii) gives the theoretical behavior of the RT and ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$ at low temperatures using the ballistic approximation (Eqs. 6 and 7). (E, i) The RT distributions (normalized by the mean RT) for the three regimes (denoted by stars in C, i). (E, ii) Typical trajectories during the decision-making process corresponding to the RT distributions shown in E, i. (E, iii) Comparison of the RT distributions of the ordered (green) and intermittent (orange) regimes with the disordered regime (black identity line) using a quantile–quantile representation.

For all regions of the phase space, the error rate and the mean RT decrease as the strength of the evidence grows (Fig. 3A, ii and iii). This happens because the positive bias increases the probability of the spin activation in the first group, and hence, the positive MF velocity increases (solution of Eq. 4), compared to the negative solution (see the asymmetric location of the green and orange vertical lines in Fig. 3A, i.

At fixed ${ϵ}_1$, we find that as we approach the transition line inside the ordered phase, the error rate decreases with increasing temperature and inhibition while the RT increases, as shown in Fig. 3. Indeed, the error rate is very low in the disordered phase and partly in the intermittent phase (Fig. 3B). However, the RT grows drastically with temperature and inhibition (Fig. 3C). This behavior of the model introduces the speed–accuracy trade-off, suggesting that an optimal range of parameters could be where the error is reasonably small while the RT is still not too large. Above the transition line, in the disordered phase, the DV in our model has a low diffusion coefficient (13), thereby giving rise to slow and accurate decisions. Below the transition line, in the ordered phase, the effective diffusion coefficient increases with decreasing temperature (approaching ballistic motion at low $T$), giving rise to fast and inaccurate decisions.

Another property that we can compare to the DDM is the ratio between the RTs of the correct (${\mathrm{RT}}_{\mathrm{c}}$) and wrong (${\mathrm{RT}}_{\mathrm{w}}$) decisions. This ratio (${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$) is equal to one for the regular DDM with symmetric thresholds (SI Appendix, section S4.C), but in the IIM, we find that there can be deviations from this strict equality (Fig. 3D; larger deviations predicted for stronger preferences are shown in SI Appendix, Fig. S7D). In Fig. 3D, ii, we plot this ratio as a function of temperature for zero inhibition and find three regimes, depending on the type of motion of the DV: ballistic (I), run-and-tumble (II), and diffusion (III).

At low temperatures (zone I in Fig. 3D, ii), the DV’s motion is ballistic (Fig. 1C). We can estimate the mean RT for the ballistic trajectories (the black curve in Fig. 3C, ii) that reached the positive or the negative threshold as

$$\begin{array}{c}\hfill R{T}_{\mathit{bal}}^{\pm }=L/|V_{\text{MF}}^{\pm }|,\end{array}$$ [6]

where $V_{\text{MF}}^{\pm }$ are the solutions of the MF equation (Eq. 4); see also SI Appendix, section S4.B for the estimation of the error rate. The ratio of these RTs (the black curve in Fig. 3D, ii) is given by

$$\begin{array}{c}\hfill \frac{{\mathrm{RT}}_{\mathrm{c}}}{{\mathrm{RT}}_{\mathrm{w}}}=\frac{|V_{\text{MF}}^{-}|}{V_{\text{MF}}^{+}}.\end{array}$$ [7]

In the biased case, the velocity toward the positive threshold $V_{\text{MF}}^{+}$ is larger than toward the negative threshold $V_{\text{MF}}^{-}$ (green and orange vertical lines in Fig. 3A, i) so that the ratio ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$ is lower than 1.

In the disordered regime of the IIM (zone III in Fig. 3D, ii), the motion of the DV is identical to the DDM (6, 7) (Fig. 1 B and D), and the RTs can be calculated analytically, giving rise to ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}=1$ (shown in SI Appendix, section S4.C).

In the ordered phase close to the second-order transition line (zone II in Fig. 3D, ii), the system’s trajectories consist of intervals of movement at nearly constant velocity, interrupted by reversals in the direction of motion, which can be approximated by the RnT motion (Fig. 1 B and F; see details in SI Appendix, section S4.D and Fig. S10). We find that the RnT motion, where the evidence is introduced by unequal flipping rates (higher flipping rate toward the correct positive threshold) while keeping the run velocity equal in both directions, results in ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}=1$. However, in the IIM, we observe a higher run velocity toward the correct decision threshold (Fig. 3A, i), and this makes ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}<1$.

At the same time, in the RnT-like regime of the IIM, tumble events have finite durations due to spin-flipping, which takes a finite time to switch the global state (SI Appendix, Fig. S10C). When we analyze the RnT motion with finite durations at each tumble event, we can obtain a RT that is longer for the correct vs. wrong decisions (SI Appendix, section S4.D and Fig. S9D). This occurs because trajectories that reach the correct threshold tend to include longer paths with more frequent tumble events that slow down the decision process. Thus, the IIM in the RnT regime (zone II in Fig. 3D, ii) exhibits ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$ that can be both smaller and larger than 1.

The final property of the IIM, which we show in Fig. 3E, is the RT distribution in the different phases, corresponding to the stars in Fig. 3C, i. Typical RT distributions are shown in Fig. 3E, i (the time is normalized by the mean RT), and typical trajectories are shown in Fig. 3E, ii. The RT distributions of the ordered and intermittent phases show sharp peaks due to the ballistic nature of the trajectories. The differences between the three distributions are quantified using the quantile–quantile plot (34, 35) (Fig. 3E, iii). The distributions for both the ordered and intermittent phases are significantly different from the disordered phase for short times, but the ordered phase close to the transition has a long-time tail similar to the disordered phase (see SI Appendix, section S4.A for further quantitative measures of these distributions).

Note that in the IIM, the dynamics depend on the system size (number of spins $N$), as shown in SI Appendix, section S2.D and Fig. S4E. However, close to the transition line, the dependence on $N$ diminishes (13), making the behavior near the transition line insensitive to fluctuations in the number of participating neurons. This is also the region which does not change its dynamics when there are fluctuations in the interaction strength between the spins (SI Appendix, section S1 and Fig. S1).

Overall, our IIM exhibits decision-making properties that are similar to the DDM with different regimes of effective diffusion coefficient as a function of our model parameters (temperature and inhibition). However, transitioning from simple diffusion to the ballistic or RnT motion in the ordered phase leads to qualitative deviations from the DDM-like behavior.

Special Properties of the IIM Near the Tricritical Point

In Fig. 3 B and C, we demonstrated that the IIM predicts a trade-off between accuracy and speed, suggesting that the region around the phase transition line allows a compromise between these conflicting traits. Since temperature $T$ represents the noise in the neural network, it may be less amenable to easy control and adjustment. In contrast, the global inhibition $\eta$ can be readily adjusted by the activity of inhibitory neurons. Motivated by this observation, we explore the role of inhibition as the control parameter that the brain adjusts to tune its accuracy, as indicated by recent experiments (12).

In Fig. 4A, we plot the points (blue) in the $T,\eta$ parameter space that have the same accuracy ($30\%$ errors) for a given constant (and small) ${ϵ}_1$ (as in Fig. 3B, i). We then consider shifting these points by increasing the inhibition by a small fixed increment $d\eta$ and analyze how the error rate and the RT change due to this shift (green circles in Fig. 4 A and B). In response to the small increase in inhibition, the error rate decreases (Fig. 4B), while the RT increases as expected (Fig. 4C). We find that the decrease in error is the most significant for lower temperatures, where the shift in inhibition moves the IIM along the sharp gradient of the error contours (Fig. 3B, i). At higher temperatures, the shift in inhibition has a vanishing effect on the accuracy, as it corresponds to moving the IIM along the error contour. This analysis suggests that the region close to the tricritical point may be advantageous with respect to allowing the brain to gain accuracy per minimal increase in inhibitory tone. Note that the exact location of the error rate minimum depends on the choice of parameters (SI Appendix, section S5.A and Fig. S11 A–C).

Fig. 4.

Inhibition controls accuracy. (A) At fixed ${ϵ}_1=0.01$, we find the points on the phase diagram that have a fixed error level of 0.3 (blue circles). The green circles denote a shift of the blue circles by increasing the global inhibition by $d\eta =0.02$. (B) The error rate and (C) the RTs are shown for the blue and green points from (A). We find that the small increase in inhibition at low temperatures leads to a significant error reduction while the RT drastically increases. At high $T$, both the error and the RT are less affected by the increase in inhibition.

In SI Appendix, section S5, we present several additional properties of the IIM near the transition line. We demonstrate that near the transition line, the IIM predicts that the errors are minimized if the decisions are forced to occur within a finite time duration (SI Appendix, section S5.B). In most realistic situations, decisions are indeed bound within some finite duration constraint.

We also show that in the region near the tricritical point it is most “costly” for the system to reach high accuracy, involving the largest relative increase in the evidence strength and neuronal activity, but on the other hand, has the advantage of giving the largest relative decrease in the RT, and the accuracy of the decisions is least sensitive to a constant rate of preference decay (SI Appendix, section S5.C).

To summarize, we find that near the transition line close to the tricritical point (in the ordered or intermittent phases), there are special properties of the IIM that may be advantageous for the decision-making process. In this region, the dynamics are most robust to fluctuations in the network size and connectivity, and it may be an optimal compromise between speed and accuracy with the ability to most significantly improve accuracy with a small increase in global inhibition. These properties suggest that the decision-making circuit in the brain may correspond in our model to the region in the vicinity of the tricritical point. In the next section, we analyze experimental data, which we compare to the IIM, and find support for this hypothesis.

Testing IIM with Experimental Data

Decision-making models are used to explain behavioral data from two-choice decision experiments, such as the accuracy and RT distributions. We compare the IIM to experimental data from two different types of decision-making tasks. The first is the well-established random-dot-motion (RDM) task, where we use published data (6, 7). The second test for our model is a two-armed bandit game (36), for which we present experimental data. One such experiment is shown here, while another is given in SI Appendix, section S7.

The RDM is a perceptual decision-making task, where the evidence is accumulated within each trial, while in the two-armed bandit game, the participants learn the hidden information about the presented stimuli (symbols or characters) over a sequence of trials. By fitting the IIM to both perceptual and reinforcement learning tasks, we demonstrate its generality, with all the experiments indicating that the decision-making circuit of the brain is in the critical regime (close to the transition line) of this model. Note that the DDM was also used to fit both types of decision-making experiments (1, 37–39).

Since in the IIM, the time is normalized in units of the spin-flip rate ($r_0$, Eq. 3), we compare it to the experimental RTs presented as ratios.

RDM Task.

The data are obtained from an RDM task (extracted from refs. 6 and 7). In this task, participants see a set of dots where a certain proportion of the dots move in one direction (either to the left or right), and the rest move in random directions. The subject’s goal is to decide whether the direction of the coherently moving dots is to the right or the left. Stimulus difficulty is varied via the proportion of dots that moved in the same direction (termed “coherence”), typically ranging from near 0 to 50% for each direction. The fewer dots that move coherently, the more challenging the task becomes.

The responses and the RTs were measured, with the following trends: More difficult conditions (low coherence) have lower accuracy (Fig. 5A) and slower responses (Fig. 5B). The correct responses (${\mathrm{RT}}_{\mathrm{c}}$, green) are the “right” responses for positive coherence, and the wrong responses (${\mathrm{RT}}_{\mathrm{w}}$, orange) are the right responses for negative coherence (Fig. 5A). The error rate represents the proportion of right responses at negative coherence and “left” responses at positive coherence. The overall mean RT (black) is calculated as a sum of ${\mathrm{RT}}_{\mathrm{c}}$ and ${\mathrm{RT}}_{\mathrm{w}}$ weighted by the response accuracy (SI Appendix, section S8 and Fig. S17A). The data show that the mean RT is higher for the higher errors (black crosses in Fig. 5E).

Fig. 5.

Fitting the IIM to the RDM task (adapted from ref. 6; see Experiment 1 in (7)). (A) Response proportion plotted against motion coherence. (B) Mean response time (RT) plotted against motion coherence. (C) Heatmap of the mean squared error (MSE), which measures the deviation of the IIM normalized mean RT as a function of the error rate from the experimental data (E). To fit the IIM, at each point in the phase diagram, we find the evidence strengths (${ϵ}_1$) that correspond to the error rates at the different coherence levels. The pink contour indicates the region of low MSE of the normalized mean RT, while the yellow contour shows the region with low MSE with respect to the mean ratio of ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$ (shown in SI Appendix, section S8 and Fig. S17C). The region that has a low MSE of these two measures (where the pink and yellow contours overlap) is indicated by the green contour. (D) The IIM’s phase diagram. The red and blue lines indicate the first- and second-order transition lines. The stars indicate the locations of parameters for which we compare the IIM to the data in (E and F). The contours are as in (C). (E) Comparison of the IIM’s prediction for the mean RT (colorful lines with circles) with the data (black line with crosses), plotted as functions of the error rate and normalized by the mean RT at the lowest error. The IIM’s parameters are shown in (D) with the same color. The colorful circles are the result of the IIM simulations over multiple trajectories, and the corresponding lines are the polynomial fit. (F) Comparison of the IIM’s prediction for the mean ratio of the RTs in the correct and wrong decisions (colorful lines with circles) with the data (black line with crosses, calculated from B), plotted as functions of the error rate.

We now fit our IIM, in the different phases, to the experimental data. In the experiments, the level of coherence is associated with the strength of the evidence that affects decision-making. Thus, we extract the values of evidence (${ϵ}_1$) that satisfy the error rates associated with each value of coherence, for each point in the phase diagram (the procedure is described in SI Appendix, section S8). Given the values of ${ϵ}_1$, we then find the corresponding mean RT, ${\mathrm{RT}}_{\mathrm{c}}$, and ${\mathrm{RT}}_{\mathrm{w}}$.

We normalize the mean RT for each error rate (coherence) by the value at the highest coherence (lowest error rate), and compare it to the data (Fig. 5E). In Fig. 5C, we plot the heatmap of the mean squared error (MSE) between the experimental data and the IIM. We find a narrow region (pink contour) where the MSE is very small, indicating good agreement between the data and the IIM.

A similar analysis of the ratio of the correct and wrong RTs (${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$), as a function of the error rate (Fig. 5F), gives us another MSE heatmap, shown in SI Appendix, Fig. S17C. The contour of low MSE from these data is shown by the yellow line in Fig. 5C. The green contour indicates the region where the IIM fits both datasets best.

In Fig. 5D–F, we demonstrate the fit of the IIM to the data for several special examples of $T$, $\eta$ values (marked with stars in Fig. 5D) that correspond to a ballistic-like regime (purple), RnT motion (green, blue, and pink), and diffusion (orange). The comparison of the normalized RT predicted by the IIM (colorful lines with circles) with the experimental data (black) is given in Fig. 5E. Similarly, we compare the IIM’s prediction for ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$ (colorful lines with circles in Fig. 5F) with the data (black line with crosses, calculated from Fig. 5B).

Overall, we find that in the regime near the tricritical point (green contour in Fig. 5C), our IIM is able to fit the ratio of the mean RTs and the ratio of correct/wrong RTs as functions of the dot-motion coherence using only evidence (${ϵ}_1$) as the control parameter. Furthermore, the ratio ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$ deviates from 1 in the IIM, as observed. We also demonstrate that the quantiles of the RT distribution can be reasonably fitted within the same region near the transition line (SI Appendix, section S8 and Fig. S17E). Note that this is achieved without changing the other model parameters (such as threshold values).

In contrast, when the DDM was fitted to the data (7), it was necessary to modify not only the evidence strength (drift) but also many other parameters of the model, including the starting point, the distance between the thresholds, the nondecision component of the RT, and the magnitude of the variability of these parameters between trials.

We view this comparison as an indication that the IIM near the transition line seems to naturally fit the behavioral data observed in experiments. It also highlights that the IIM can explain the observations with fewer manipulations of the model parameters compared to the DDM.

Two-Armed Bandit Task.

Next, we analyze recent experimental data obtained from volunteers playing a two-armed bandit game. In this game, the subject chooses one option (in the form of a special character) per trial. This character either gives monetary gain (0 or $+$1) or loss (0 or $-$1) as a reward, with some fixed probabilities which are unknown to the subject (SI Appendix, Fig. S15A). The goal is to maximize the total score. Each pair of symbols refers to either gain or loss trials. We define the correct option as the option that increases the total score with a higher probability in the gain condition and decreases the total score with a lower probability in the loss condition. The gain and loss trials can be separated, so the participants learn the hidden probabilities of the rewards for one pair in a game, or alternatively, the gain and loss trials can be intermixed, and the two pairs alternate randomly in the same game. During these experiments, the choices of the participants and the corresponding decision times (RT) were registered. Since in the IIM, the time is normalized in units of the spin-flip rate ($r_0$, Eq. 3), we compare our model to the experimental RT ratios.

Since the probabilities encoded by the symbols are unknown to the participants, the initial trials give rise to a learning process during which the participants form a preference toward one of the options in each pair. Therefore, we compare our IIM to the experimental data only after this learning period is finished and the participants’ performance saturates. However, in SI Appendix, section S9, we demonstrate that our model can also be used to account for a learning process.

In SI Appendix, section S7, we present experimental data and its comparison to the IIM for a two-armed bandit game with intermixed gain and loss trials. Here, we compare the IIM to an experiment where the volunteers played separate games of gain and loss. The main novelty was the ability to monitor the concentration of inhibitory neurotransmitter GABA ($\gamma$-aminobutyric-acid) during the decision-making process. The GABA concentration was quantified from the dorsal anterior cingulate cortex (dACC), using Proton Magnetic Resonance Spectroscopy (¹H-MRS) at 7T (see more details in SI Appendix, section S10 and ref. 40).

In the experiment, 107 volunteers played four separate games (of 50 trials) in each of the combinations: gain and loss with probabilities of 65/35 and 50/50. All participants provided written informed consent prior to participation. The experiment was approved by the Wolfson Medical Center Helsinki Committee (Protocol: 0084-19-WOMC) and the Weizmann Institutional Review Board (Protocol: 1656-2).

Under the unbiased conditions, when the probabilities of zero and nonzero rewards for both options were 50% (no correct option), the RT was measured as the baseline (labeled ${\mathrm{RT}}_{50/50}$). Following the initial learning period (28 trials), trials 29 to 50 are used for the data analysis. The gain and loss trials did not show significant differences in their error rates and RT, so their data were combined (SI Appendix, section S10.B).

For a further analysis, we divided the participants into three groups according to their error rates and RTs normalized by the unbiased RTs (i.e., ${\mathrm{RT}}_{65/35}/{\mathrm{RT}}_{50/50}$, Fig. 6A, i). The orange group indicates the volunteers who did not learn the correct choice very well and had an error rate larger than $0.15$ (an arbitrary threshold, but the analysis is insensitive to the value of this threshold as shown in SI Appendix, section S10.C).

Fig. 6.

Two-armed bandit task (separate gain and loss trials) compared to the IIM. (A, i) Normalized RT (${\mathrm{RT}}_{65/35}/{\mathrm{RT}}_{50/50}$) as a function of the error rate. Each mean RT in the gain or loss trials (with the probabilities of 65 to 35% per choice) is normalized by the participant’s RT in the unbiased condition (with the reward probability of 50% per choice). Each point represents a result from a single participant in the gain and loss games separately. The point size is related to the average GABA concentration measured for each participant during the task. The data are divided into three groups: the green group has an error rate $\le$ 0.15 and ${\mathrm{RT}}_{65/35}/{\mathrm{RT}}_{50/50}\le 1$, the blue group has an error rate $\le$ 0.15 and ${\mathrm{RT}}_{65/35}/{\mathrm{RT}}_{50/50}>1$, while the orange group has an error rate $>$ 0.15. The vertical and horizontal lines denote the average error rate and normalized RT for each group. (A, ii) The GABA concentration quantified from the dACC during the task for the unbiased and biased trials for the green and blue groups. We find that the concentration of ${\mathrm{GABA}}_{50/50}$ (green) is not different from ${\mathrm{GABA}}_{50/50}$ (blue) (unequal variance t test: $P=0.74$). However, in the biased conditions, the concentration ${\mathrm{GABA}}_{65/35}$ (green) shows a difference with ${\mathrm{GABA}}_{65/35}$ (blue) (unequal variance t test: $P=0.0375$). (A, iii) The ratio ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$ for the same groups of A, i at the biased conditions (both gain and loss). Each point represents a result from a single participant. The size of the points is related to the average GABA concentration of each participant during the task. The blue line indicates a 4th-degree polynomial fitted to all the data points, as a guide to the eye. (B, i) The normalized RT ratio (between the biased and unbiased conditions) for the average error rate of the green group (Table 1), as given by the IIM. For each point of the phase diagram, we find the values of ${ϵ}_1$ that satisfy the error rate of the green group ($0.06\pm 0.01$), and this gives us the biased ${\mathrm{RT}}_{65/35}$. The dark green contour denotes the region that fits the green group’s normalized RT ratio (${\mathrm{RT}}_{65/35}/{\mathrm{RT}}_{50/50}$, without constraining the ratio ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$). The green circles correspond to values of the parameters that also fit the green group’s ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$ ratio. The blue circles denote a shift of the green circles by increasing the global inhibition by a factor of $1.17$, which is the ratio of the measured average GABA concentrations in the two groups for the biased conditions (in A, ii). The red and blue lines on the heatmaps denote the first- and second-order transitions, respectively. (B, ii) Heatmap of the ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$ given by the IIM for the average error rate of the green group under the biased conditions. (B, iii) Comparison of the phase space regions that fit the data in two versions of two-armed bandit experiments. The turquoise contour indicates the area of the phase space that best matched the data of the experiment with intermixed gain/loss trials in SI Appendix, section S7 and Fig. S15 B and C. (C, i and ii) The error rate and preferences of the green and blue groups from B, as a function of temperature $T$. The calculated error rates agree with the mean values of the experimental observations (denoted by the horizontal lines and shading). (C, iii) The ratio ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$ for the green and blue circles in B. The green circles agree well with the experimental observation (denoted by the horizontal lines and shading), while the blue circles indicate a lower agreement. (C, iv) The ratio of the mean RTs for the green and blue circles in B as a function of temperature $T$. The black line and the shaded area indicate the ratio of the average biased RTs between the blue and green groups in the experiment: $1.33\pm 0.20$. The purple box indicates the region of best agreement near the tricritical point (dashed vertical line denotes $T_{\text{tri}}$).

The error rate of the green group is below the $0.15$ threshold, and the ratio ${\mathrm{RT}}_{65/35}/{\mathrm{RT}}_{50/50}\le 1$. This group of participants makes accurate and fast decisions, as expected in our model, since a strong preference gives rise to accurate decisions and decreases the RT (Fig. 3A, ii and iii). This general property also appears in the DDM.

Surprisingly, we find another group of participants (blue group, Fig. 6A, i) that make accurate decisions (error rate $\le$ 0.15), but slower in the biased case than in the unbiased one: the ratio ${\mathrm{RT}}_{65/35}/{\mathrm{RT}}_{50/50}>1$. The low error rate indicates a significant preference, and it is, therefore, surprising that this does not manifest in faster decisions. An important observation that may explain this puzzling group of volunteers is given in Fig. 6A, ii. For the green and blue groups, we compare the GABA concentration during the tasks in the biased and unbiased conditions. We find that the GABA concentrations in both groups are not different in the unbiased case, while in the biased conditions, the blue group exhibits larger concentrations compared to the green group (the statistical analysis is shown in SI Appendix, Table S8).

Finally, we plot the ratio of correct vs. wrong RTs for the three groups (${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$, Fig. 6A, iii). The experimental data show that at very low error rates, this ratio for the green and blue groups combined significantly drops below 1 (blue line in Fig. 6A, iii, and SI Appendix, Table S8), indicating a deviation from the DDM prediction (where this ratio is strictly 1).

We now systematically compare all of the experimental data described above and summarized in Table 1 to our IIM. We start by fitting the data of the green group. For each point ($T$, $\eta$), we find the values of ${ϵ}_1$ corresponding to the measured average error rate of $0.06\pm 0.01$. Using these values, we derive the normalized RT (${\mathrm{RT}}_{65/35}/{\mathrm{RT}}_{50/50}$, Fig. 6B, i) and the ratio ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$ (Fig. 6B, ii). Then, we first select the region that fits the observed normalized RT of $0.81\pm 0.02$, denoted by the dark-green contour. Next, we also fit to the observed ratio ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}=0.91\pm 0.07$ and find a narrow region near the tricritical point, denoted by the green circles in Fig. 6B (see more details in SI Appendix, sections S6.B and S10.C). It is satisfying to find that this narrow region of parameters lies close to the edge of the region that fits the version of the experiment with intermixed gain/loss trials (turquoise contour in Fig. 6B, iii; SI Appendix, section S7) and the RDM experiment (Fig. 5C).

Table 1.

Experimental data from the two-armed bandit task with separate gain and loss trials

Mean $\pm$ SE	Green	Blue	Orange
Size	21	17	106
Error rate	$0.06\pm 0.01$	$0.06\pm 0.01$	$0.31\pm 0.01$
${\mathrm{RT}}_{65/35}$ (s)	$0.59\pm 0.05$	$0.79\pm 0.06$	$0.68\pm 0.02$
${\mathrm{RT}}_{65/35}/{\mathrm{RT}}_{50/50}$	$0.81\pm 0.02$	$1.33\pm 0.08$	$1.00\pm 0.02$
${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$, 65/35	$0.91\pm 0.07$	$0.86\pm 0.09$	$0.98\pm 0.02$
${\mathrm{GABA}}_{65/35}$	$1.00\pm 0.04$	$1.17\pm 0.07$	$1.03\pm 0.02$
${\mathrm{GABA}}_{50/50}$	$1.05\pm 0.05$	$1.08\pm 0.06$	$1.04\pm 0.02$

Experimental results, calculated for 107 volunteers, participated in four games of 50 trials of two-choice tasks under uncertainty. In two games, the symbols in each pair encode a monetary gain or loss with 65 and 35% probabilities, and in the other two games, the symbols give a reward with equal probabilities of 50%. The results of the gain and loss trials are combined, as they do not show any significant differences (SI Appendix, section S10). The group names relate to the color in Fig. 6A, i. The error rate indicates the proportion of the wrong choices in trials 29 to 50 after the learning period. The mean RTs are measured during the trials after the learning period, and the ${\mathrm{RT}}_{\mathrm{c}}$ and ${\mathrm{RT}}_{\mathrm{w}}$ are the mean RTs in the correct and wrong decisions. The GABA concentration is quantified from the dACC during the task for the biased and unbiased conditions. The ratio of the GABA concentrations in the blue and green group is ${\mathrm{GABA}}_{65/35}$(blue)/${\mathrm{GABA}}_{65/35}$(green) = $1.17\pm 0.11$. The ratio of the RTs in the same groups is ${\mathrm{RT}}_{65/35}$(blue)/${\mathrm{RT}}_{65/35}$(green) = $1.33\pm 0.20$.

Next, we wish to explain the behavior of the blue group using our model. Guided by the observation of larger inhibitory signals for these volunteers compared to the green group (Fig. 6A, ii), we assume a simple linear relation between the GABA concentration and the level of global inhibition in the IIM. We, therefore, use the measured ratio ${\mathrm{GABA}}_{65/35}$(blue)/${\mathrm{GABA}}_{65/35}$(green) $=1.17\pm 0.11$ (Table 1) to place the blue group above the green group at larger values of $\eta$ (the blue circles in Fig. 6B). Note that an increase in the cross-inhibition between the spin groups, which will also manifest in an increase in the GABA concentration, has the opposite effect of lowering the accuracy and decreasing the RT (SI Appendix, section S2.C and Fig. S4D).

We now wish to test whether our interpretation of the blue group as having higher global inhibition is consistent with the observed increase of the ${\mathrm{RT}}_{65/35}$ between the green and blue groups (Table 1). We start by finding the values of ${ϵ}_1$ in the IIM for the blue points (Fig. 6C, ii), which give us the observed error rate of $0.06\pm 0.01$ (Fig. 6C, i). Using these values, we extract the ratio ${\mathrm{RT}}_{\mathrm{c}}/{\mathrm{RT}}_{\mathrm{w}}$ (Fig. 6C, iii), and we see that the blue points are slightly outside the experimental data. We next calculate the biased RTs (${\mathrm{RT}}_{65/35}$) and their ratio ${\mathrm{RT}}_{65/35}$(blue)$/{\mathrm{RT}}_{65/35}$(green) (Fig. 6C, iv). We find that the RT ratio calculated from our IIM agrees very well with the observed value ($1.33\pm 0.20$; black line and shading) at temperatures close to the tricritical point (purple contour in Fig. 6B, iii).

The comparison between the experimental data and the analytic relation between error and normalized RT for the DDM shows that this model does not agree with the data (SI Appendix, section S10.D and Fig. S23B).

The good agreement between the IIM and the data suggests the following interpretation of the differences between the two groups of volunteers. While the green group learns the correct choice with a strong preference (therefore, having fast and accurate decisions), the blue group does not learn so well (weaker preference, blue in Fig. 6C, ii) but instead compensates with an increase in inhibition (exhibiting higher GABA concentrations) to improve the accuracy of the decisions. Without the increase in inhibition, these weaker preferences would result in much higher errors (SI Appendix, section S10.C and Fig. S23A). The price that the blue group pays is slower decisions compared to the green group. The self-consistency and robustness of our analysis and agreement with the IIM are demonstrated using different threshold values to divide the data into the three groups (SI Appendix, section S10.C and Figs. S21 and S22).

Discussion

We have presented here a theoretical framework for describing the decision-making process in the brain, the IIM. It is based on Ising spins whose state represents the firing of neurons, arranged in groups that represent each of the available options, and interact in an excitatory manner within the group while cross-inhibiting the spins in the other group. The states of these spins drive the changes in the state of an integrator, which acts as the decision variable (DV), and upon reaching one of two threshold values, a decision is made. This last property is identical to the highly successful drift-diffusion model (DDM) (6), which the IIM recovers in its disordered regime (high levels of noise and inhibition). The IIM incorporates global inhibition as a biologically inspired mechanism in the Ising-type dynamics and provides clear predictions that can be tested in experimental data.

The IIM model goes beyond the DDM, and in its ordered phase, it displays run-and-tumble (RnT) dynamics with significant deviations from the DDM. In this phase, we find faster and more accurate decisions that may be optimized near the 2nd-order phase transition line. Just below this transition line near the tricritical point, the model predicts maximal gain in accuracy as a function of an increase in global inhibition, suggesting a mechanism for explaining the observed increased inhibition when the task is more difficult (12). In the same region below the transition line, we find a minimum in the rate of accuracy decay per loss of learned preference, which is an advantage for maintaining accuracy for a longer time. This region is also where the behavior is insensitive to fluctuations in the size of the spin groups and the strength of their interactions. Therefore, the IIM suggests that it can be advantageous for the brain to be in this critical region.

We test our IIM with two types of decision-making task experiments, which support the model’s prediction regarding the importance of the critical regime. The first is the perceptual RDM experiment (6, 7). Comparing the IIM to the RDM task shows that it is able to fit the experimental data near the transition line, using fewer control parameters compared to the DDM.

The second type of experiments is the two-armed bandit task, where the decision-making is based on reinforcement learning, from two new experimental datasets. The intermixed gain/loss experiment (SI Appendix, section S7) allows us to map the data to a region of the IIM phase space ($T$, $\eta$) around the tricritical point and the 2nd-order transition line. The separate gain/loss experiment also contained measurements of changes in the inhibition strength within the decision-making region of the brain. This dataset again localizes the area of the IIM phase space that fits the data in the vicinity of the tricritical point. Notably, the standard DDM does not provide a good fit for both experimental datasets.

The IIM’s ability to effectively explain the data from both perceptual decision-making (RDM) and reinforcement learning (two-armed bandit) tasks, both of which constrain the model to operate in the critical regime near the transition line (SI Appendix, Fig. S23C), demonstrates the generality of this model.

Using our model, we interpret the experimental data as indicative of two ways the brain can make accurate decisions. The first is based on developing a strong preference for the correct choice, leading to fast decisions, while the second is based on a weaker preference, compensated by higher inhibition that leads to slow and accurate decisions. These two processes may be reminiscent of the fast-and-slow decision-making processes described by D. Kahneman’s “Thinking, Fast and Slow” (41).

These experimental results, therefore, suggest that the brain utilizes the special properties of the critical region near the phase transition line. This result is different from the criticality that was proposed to exist in the brain with respect to the structure and the spatial connectivity of the neural network (42–45), including the effects of global inhibition (46), or in collective animal systems (47). It is another form of criticality that is not manifested in the spatial interactions (which are all-to-all in our model) but instead in the dynamics of the decision-making process.

Our findings indicate that the model operates close to the transition line near the tricritical point, suggesting that the neural circuit involved in decision-making is experiencing strong inhibition. Within our IIM, a situation where strong evidence is presented for both options is equivalent to shifting to a lower global inhibition. Under such conditions, we expect that the decisions would become faster and less accurate (Fig. 3 B and C). This allows us to naturally explain the following observations. The first is the “magnitude effect,” where strong evidence is provided for both choices (48), resulting in faster and less accurate decisions. Similarly, in ref. 49, it was found that introducing more background odors while maintaining the strength of the target odorant resulted in shorter RT and increased error rate, which is equivalent to a global decrease in inhibition in the IIM. This explanation is different from the DDM, where such behavior is attributed to higher effective noise, which is less related to the increase in the external strength of the evidence.

Finally, when the decision-making circuit operates near the tricritical point in our model, a reduction in global inhibition can serve as a mechanism for forcing a decision, similar to invoking a time-dependent decrease of the decision thresholds (50). Indeed, there is experimental evidence for an increase in global neuronal firing activity during temporally constrained decision-making processes (51). The IIM, therefore, provides an explanation for how the brain might use modulation in the global inhibition to modify the decision performance (RT and error rate).

Note that our spin-based model for decision-making was motivated by the success of the spin model in describing the decision-making of individual animals and animal groups while navigating through space (14, 15, 17). More theoretical work is planned to further elucidate the properties of the IIM, such as extending it to describe more than binary choices (4, 34, 52, 53), and to explore the IIM in connection with spatial navigation. In addition, future experimental work is needed to further test the predictions made in this work regarding the criticality and the run-and-tumble nature of the neuronal dynamics during decision-making.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Anonymized behavioral data and GABA measurements from the experiments (anonymous), scripts for the data analysis, and scripts for the model simulations have been deposited in github/zenodo (https://doi.org/10.5281/zenodo.16498152). Previously published data were used for this work. Two-armed bandit task: in our manuscript, we use the processed data (averaged GABA measurements and behavioral data). The raw experimental data and the corresponding protocols and procedures are fully described in Finkelman et al. (40). Random-dot motion task: Fig. 5 A and B in the main text and SI Appendix, Fig. S17A contain adapted data from Ratcliff et al. (6) (figure 3) and Ratcliff and McKoon (7) (Experiment 1, figure 7).