Astrocytes Learn to Detect and Signal Deviations From Critical Brain Dynamics

Abstract

Astrocytes are nonneuronal brain cells that were recently shown to actively communicate with neurons and are implicated in memory, learning, and regulation of cognitive states. Interestingly, these information processing functions are also closely linked to the brain's ability to self-organize at a critical phase transition. Investigating the mechanistic link between astrocytes and critical brain dynamics remains beyond the reach of cellular experiments, but it becomes increasingly approachable through computational studies. We developed a biologically plausible computational model of astrocytes to analyze how astrocyte calcium waves can respond to changes in underlying network dynamics. Our results suggest that astrocytes detect synaptic activity and signal directional changes in neuronal network dynamics using the frequency of their calcium waves. We show that this function may be facilitated by receptor scaling plasticity by enabling astrocytes to learn the approximate information content of input synaptic activity. This resulted in a computationally simple, information-theoretic model, which we demonstrate replicating the signaling functionality of the biophysical astrocyte model with receptor scaling. Our findings provide several experimentally testable hypotheses that offer insight into the regulatory role of astrocytes in brain information processing.

1. Introduction

Long thought to only provide structural and metabolic support for neuronal networks (Tsacopoulos & Magistretti, 1996), astrocytes are ubiquitous nonneuronal brain cells that are now known to form a complementary information processing layer that actively communicates with neurons at multiple spatiotemporal scales (Dallérac & Rouach, 2016; Halassa & Haydon, 2010; Perea, Sur, & Araque, 2014). Indeed, astrocytes interface with individual synapses (Chen et al., 2013; Perea & Araque, 2010), single neurons (Letellier et al., 2016; Parpura & Haydon, 2000), and neuronal networks (Pascual et al., 2005; Perea, Yang, Boyden, & Sur, 2014) and are ultimately linked to behavior (Bojarskaite et al., 2020; Brancaccio, Patton, Chesham, Maywood, & Hastings, 2017; Foley et al., 2017; Halassa & Haydon, 2010; Mu et al., 2019). This newfound symbiosis between astrocytes and neurons challenges the long-standing neuron doctrine (De Pittà & Berry, 2019) by giving rise to new hypotheses that explain how mechanisms of neuron-astrocyte interaction lead to the functional role astrocytes play in motor behavior (Merten, Folk, Duarte, & Nimmerjahn, 2021), cognitive states (Santello, Toni, & Volterra, 2019), memory (Kol et al., 2020; Lee et al., 2014), and learning (Stewart, Padmashri, Suresh, Boska, & Dunaevsky, 2015).

Mounting empirical and theoretical evidence has also attributed the brain's impressive information processing capabilities, including memory and learning (Bertschinger & Natschläger, 2004; Legenstein & Maass, 2007; Meisel, Klaus, Vyazovskiy, & Plenz, 2017; Shew et al., 2011), to its operation near a critical phase transition (Bak, Tang, & Wiesenfeld, 1987; Plenz et al., 2021). This narrow boundary separates two phases of activity: subcritical dynamics, where stable network activity supports memory function, and supercritical dynamics, where chaotic activity has been attributed to learning (Li et al., 2019; Skilling, Ognjanovski, Aton, & Zochowski, 2019). Combining the computational properties of both memory and learning, networks poised near the critical phase transition are well known to maximize their information processing capacity (Bertschinger & Natschläger, 2004; Legenstein & Maass, 2007; Shew, Yang, Yu, Roy, & Plenz, 2011). Indeed, significant deviation into either phase has been associated with dysfunctional brain activity (Meisel, Storch, Hallmeyer-Elgner, Bullmore, & Gross, 2012; Montez et al., 2009), while the continuum of dynamics around the vicinity of the critical phase transition has been attributed to a range of behavioral states (Bellay, Klaus, Seshadri, & Plenz, 2015; Fagerholm et al., 2015; Hahn et al., 2017; Priesemann, Valderrama, Wibral, & Le Van Quyen, 2013; Tagliazucchi et al., 2016). Interestingly, some of these behavioral states have also been linked to astrocyte calcium ion concentration $\left[{\mathrm{Ca}}^{2+}\right]$ signaling (Bojarskaite et al., 2020; Foley et al., 2017; Halassa & Haydon, 2010; Ingiosi et al., 2020; Thrane et al., 2012). While waves of intercellular, $\left[{\mathrm{Ca}}^{2+}\right]$, have been extensively studied in astrocytes, a mechanistic understanding of their role in brain information processing remains elusive (Guerra-Gomes, Sousa, Pinto, & Oliveira, 2018; Semyanov, Henneberger, & Agarwal, 2020; Semyanov & Verkhratsky, 2021).

Astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ waves are a central signaling mechanism involved in neuron-astrocyte communication (De Pittà & Berry, 2019) that enables astrocytes to both integrate neuronal synaptic activity (Schummers, Yu, & Sur, 2008; Wang et al., 2006; Zhang, Chen et al., 2016) and, in turn, act on the presynaptic and postsynaptic sites through a multitude of biochemical pathways (De Pittà, 2020). Forming tripartite connections with tens of thousands of synapses (Zhou, Zuo, & Jiang, 2019), astrocytes integrate extracellular glutamate that is released from presynaptic terminals, as a result of neuronal activity, into somatic $\left[{\mathrm{Ca}}^{2+}\right]$ waves that operate at a timescale of seconds (Araque et al., 2014; Perea, Sur et al., 2014; Wang et al., 2006). Interestingly, this signaling pathway was recently found to exhibit bidirectional scaling of astrocyte glutamate receptor expression in response to changes in neuronal activity (Xie et al., 2012). Specifically, suppression of neuron action potentials for 4 to 6 hours was found to produce significantly faster somatic $\left[{\mathrm{Ca}}^{2+}\right]$ rise time in astrocytes, while a 4 to 6 hour increase in neuronal firing rates resulted in a slower $\left[{\mathrm{Ca}}^{2+}\right]$ rise time. These unique spatiotemporal features of astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ signaling combined with its connections to cognitive functions including memory, learning, behavioral states, and, in turn, critical network dynamics, lead to our hypothesis that astrocytes can conveniently use $\left[{\mathrm{Ca}}^{2+}\right]$ signaling as a proxy for brain activity and, in turn, signal when network activity deviates from critical dynamics by rescaling of glutamate receptor expression, thereby acting as an early warning system of disruptive dynamics for information processing.

Here, we computationally investigated the functional role of astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ waves with respect to critical brain dynamics. To do so, we developed a biologically plausible neuronal network that served as a model of presynaptic activity combined with a biophysical astrocyte model to suggest how, and to what extent, astrocytes are sensitive to changes in the dynamics of presynaptic activity. We demonstrate how astrocytes use $\left[{\mathrm{Ca}}^{2+}\right]$ waves to signal short-term (on the order of seconds) deviations into either supercritical or subcritical dynamics by initially learning the information content of near-critical presynaptic activity that spans several hours. This resulted in a computationally simple, information-theoretic astrocyte model that reproduced the biophysical model's $\left[{\mathrm{Ca}}^{2+}\right]$ signaling functionality. By showing how astrocytes can learn to detect and signal network deviations away from near-critical dynamics, our model suggests a new regulatory role for astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ wave signaling with implications for brain information processing.

2. Methods

To investigate how changes in neuronal network dynamics may have an impact on astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ signaling, we developed a biologically constrained model where astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ is driven by neuronal presynaptic activity. We modeled neuronal activity as a two-dimensional slice of neural tissue with activity patterns mimicking presynaptic activity as would be perceived by an astrocyte. In turn, a biophysical astrocyte model integrates the presynaptic inputs. The model components are detailed below.

2.1. A 2D Model of Neural Activity with a Critical Phase Transition

We modeled the brain's neural activity as an attractor neural network that reproduces experimentally observed features accessible to astrocytes through their tripartite synaptic connections. Specifically, we modeled the production of a spike by neuron $i$ at time $t$ in ms as a binary variable ${\sigma }_i\left(t\right)=1$ and lack of a spike as ${\sigma }_i\left(t\right)=-1$. Given that astrocytes integrate neuronal presynaptic activity from tens of thousands of synapses (Vasile, Dossi, & Rouach, 2017) in a spatially localized manner (Oberheim, Wang, Goldman, & Nedergaard, 2006), where synaptic activity is known to exhibit spatially synchronous clusters (Gökçe, Bonhoeffer, & Scheuss, 2016; Kleindienst, Winnubst, Roth-Alpermann, Bonhoeffer, & Lohmann, 2011; Takahashi et al., 2012), we designed a two-dimensional lattice neural network that consists of $n=256\times 256=65,536$ neurons with nearest-neighbor interactions (see Figure 1A). Treating each neuron in the lattice as a presynaptic site at a tripartite synapse that is integrated by an astrocyte, we equated this lattice network to a two-dimensional slice of neural tissue with an area of 233 $\mu {\mathrm{m}}^2$ and thickness 0.9 $\mu \mathrm{m}$, given that cortical synaptic density is approximately 1.1 $\mu {\mathrm{m}}^{-3}$ (DeFelipe, Alonso-Nanclares, & Arellano, 2002).

Figure 1:

Neuronal network model with a critical phase transition. Presynaptic activity is modeled as an attractor neuronal network consisting of a (A) two-dimensional lattice with binary $256\times 256$ neurons that are (B) updated every 50 ms resulting in a maximum neuron spike frequency of 20 Hz. (C) Bidirectional couplings group neurons into clusters of highly coupled neurons separated by weakly coupled neurons. (D) Examples of neuronal presynaptic spike frequencies (for 135 seconds) at (left) subcritical ($T=1.7<T_c$), (middle) near-critical ($T=2.3\approx T_c$), and (right) supercritical ($T=3.3>T_c$) dynamics, where parameter $T$ sets network dynamics. Each pixel represents a single neuron/presynaptic site on the two-dimensional lattice. (E) Each network's $T_c$ is approximated by measuring the average fluctuation in (left) average network state and (right) network energy, both of which peak at $T_c$. Averaged over four realizations of the network, $T_c\approx 2.3$ ranging in $2.275\le T_c\le 2.325$.

We replicated cortical firing rates by updating the binary state of each neuron once every 50 ms (see Figure 1B), thereby resulting in neuronal spike rates that range from 0.02 to 20 Hz (O'Connor, Peron, Huber, & Svoboda, 2010; Roxin, Brunel, Hansel, Mongillo, & Vreeswijk, 2011). The probability of a given network configuration at $t$ is expressed by

$$P_t\left(\sigma \right)=\frac{1}{{\sum }_{\sigma }exp\left(-{\sum }_{i,j}{J}_{ij}{\sigma }_i\left(t\right){\sigma }_j\left(t\right)\right)}exp\left(-\sum _{i,j}{J}_{ij}{\sigma }_i\left(t\right){\sigma }_j\left(t\right)\right),$$ (2.1)

where coupling parameter $J_{ij}$ captures the interaction between neighboring neurons $i$ and $j$. This formulation is equivalent to the well-known Ising model with no external field, which has been extensively used to model experimentally observed neural activity (Kitzbichler, Smith, Christensen, & Bullmore, 2009; Schneidman, Berry, Segev, & Bialek, 2006). We updated neurons in the network according to the standard procedure for the 2D Ising model (Yeomans, 2010), flipping the binary state of each neuron $i$ from ${\sigma }_i\left(t\right)$ to ${\sigma }_i(t+1)$ with probability

$$p({\sigma }_i\left(t\right)\to {\sigma }_i(t+1))=\left\{\begin{array}{cc}exp\left(\frac{-\Delta E}{T}\right)\hfill & \text{if}\Delta E\ge 0\hfill \\ 1.0\hfill & \text{if}\text{otherwise},\hfill \end{array}\right)$$ (2.2)

where $T$ is the scalar temperature parameter that controls network dynamics (see section 2.1.2) and $\Delta E=H(t+1)-H\left(t\right)$ is the change in the network's energy, in turn defined as

$$H\left(t\right)=-\sum _{i,j}{J}_{ij}{\sigma }_i\left(t\right){\sigma }_j\left(t\right).$$ (2.3)

Once all neuron states have been updated at time $t$, we regarded each ${\sigma }_i\left(t\right)$ as a single presynaptic input to the astrocyte with corresponding extracellular glutamate concentration, ${\gamma }_i\left(t\right)$, as follows:

$${\gamma }_i\left(t\right)=\left\{\begin{array}{cc}0.2\mu \mathrm{M}\hfill & \text{if}{\sigma }_i\left(t\right)=1\hfill \\ 0.0\mu \mathrm{M}\hfill & \text{if}\text{otherwise}.\hfill \end{array}\right)$$ (2.4)

In this fashion, each time a neuron becomes active, it also releases a quantum of glutamate that we modeled as a square pulse with amplitude ${\gamma }_i\left(t\right)$ and duration 1 ms. In turn, this glutamate quantum stimulates astrocyte receptors (see section 2.2).

2.1.1. Spatial Network Model

We mimicked the experimentally observed, spatially synchronous synaptic clusters (Gökçe et al., 2016; Kleindienst et al., 2011; Takahashi et al., 2012) by grouping neurons into clusters of strong coupling separated by narrow regions of weak coupling (see Figure 1C). In agreement with experimental data, our model produces synchronous clusters ranging in size from approximately 5 $\mu {\mathrm{m}}^2$ to 30 $\mu {\mathrm{m}}^2$ (Kleindienst et al., 2011) (see Figure 1D, middle).

Clustered couplings were implemented by a randomly generated radial basis function $f_{RB}:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$, defined as

$$f_{RB}(x,y)=\frac{1}{{\sum }_{k=1}^K\left|a_k\right|}\sum _{k=1}^K{a}_{k}exp\left(\frac{-{(x-x_k^{\prime })}^2-{(y-y_k^{\prime })}^2}{d_k^2}\right),$$ (2.5)

where each radial basis $k$ is set by parameters $x_k^{\prime }$, $y_k^{\prime }$, $a_k$, $d_k\in \mathbb{R}$ (see Table 1). Next, we used $f_{RB}$ to assign preliminary coupling, $J_{ij}^{\prime }$, between neighboring neurons $i$ and $j$ through the following relation,

Table 1:

Parameters for Generating Neuron Couplings.

Symbol	Description	Value	Units
${(x^{\prime },y^{\prime })}_k$	2D lattice location of radial basis $k$	$\left(\mathbb{U}\right(0,256),\mathbb{U}(0,256\left)\right)$	1 lattice site
$a_k$	Amplitude of radial basis $k$	$\mathbb{N}(0.1,1)$	–
$d_k$	Radius of radial basis $k$	$\mathbb{N}(6,1)$	1 lattice site
$K$	Total number of radial basis functions	500	–
${\eta }_{+}$	Scaling factor	4.0	–
${\eta }_{-}$	Scaling factor	0.35	–

$$J_{ij}^{\prime }={\left(\frac{q_{ij}-q_{max}}{\overline{q}-q_{max}}\right)}^{10}+1,$$ (2.6)

with the absolute difference $q_{ij}$ defined as

$$q_{ij}=|f_{RB}(x_i,y_i)-f_{RB}(x_j,y_j)|,$$ (2.7)

where $(x_i,y_i)$, $(x_j,y_j)$ are the lattice coordinates of respective neurons $i$, $j$ and values $\overline{q}$, $q_{max}$ are, respectively, the average and maximum of $q_{ij}$ taken over all neighboring neurons.

The final coupling $J_{ij}$ was computed through a series of transformations (see equations 2.8 to 2.10) on $J_{ij}^{\prime }$,

$$J_{ij}^{\prime \prime }=\frac{J_{ij}^{\prime }}{{\overline{J}}_{ij}^{\prime }}-2$$ (2.8)

$$J_{ij}^{\prime \prime \prime }={\eta }_{\pm }{J}_{ij}^{\prime \prime }$$ (2.9)

$$J_{ij}=|J_{ij}^{\prime \prime \prime }-(1+{\overline{J}}_{ij}^{\prime \prime \prime })|$$ (2.10)

where scaling factor ${\eta }_{+}$ is used if $J_{ij}^{\prime \prime }>0$ or ${\eta }_{-}$ if $J_{ij}^{\prime \prime }<0$, and averages ${\overline{J}}_{ij}^{\prime }$ and ${\overline{J}}_{ij}^{\prime \prime \prime }$ are defined as

$${\overline{J}}_{ij}^{\prime }=\frac{1}{m}\sum _{i,j}{J}_{ij}^{\prime }{\overline{J}}_{ij}^{\prime \prime \prime }=\frac{1}{m}\sum _{i,j}{J}_{ij}^{\prime \prime \prime }$$ (2.11)

where $m$ is the total number of lattice couplings. Final couplings $J_{ij}$ range approximately from 0.5 to 1.5 with average coupling ${\overline{J}}_{ij}\approx 1.0$ (see Figure 1C).

2.1.2. Critical Phase Transition

The 2D Ising model has been demonstrated as a maximum entropy network model with near-critical dynamics that reproduces multiple features observed in brain dynamics (Fraiman, Balenzuela, Foss, & Chialvo, 2009; Kitzbichler et al., 2009). In particular, there exists a critical temperature, $T_c$, that separates two phases of dynamics: subcritical $T<T_c$ where progressively smaller $T$ produce larger clusters of synchronized neurons as illustrated in Figure 1D, (left), and supercritical $T>T_c$ where progressively larger $T$ produce increasingly decoupled random neuron activity, shown in Figure 1D, (right). We interpreted parameter $T$ as a scalar abstraction of the large number of parameters believed to have an impact on brain dynamics (Chialvo, 2010).

Since analytical solutions for $T_c$ exist only for the 2D Ising model with uniform coupling, $J=1$, we approximated $T_c$ of each custom coupled Ising neuron network (see section 2.1.1) by evaluating the fluctuation in average network state, $S\left(t\right)=\frac{1}{n}{\sum }_{i=1}^n{\sigma }_i\left(t\right)$, through expression

$$\chi =\frac{1}{T}\langle {(S\left(t\right)-\langle S\left(t\right)\rangle )}^2\rangle ,$$ (2.12)

where $\langle \dots \rangle$ is the temporal average. We also considered the fluctuation in network energy, equation 2.3, using

$$C_H=\frac{1}{T^2}\langle {(H\left(t\right)-\langle H\left(t\right)\rangle )}^2\rangle ,$$ (2.13)

For each model realization, we randomly initialized neuron states and simulated each network for 25 million iterations to reach stable dynamics, at which point we sampled $S\left(t\right)$ and $H\left(t\right)$ to evaluate $\chi$ and $C_H$ (see Figure 1E and sections 2.1 and 2.3). Since fluctuations peak near the critical transition (Yeomans, 2010), the maximum values for $\chi$ and $C_H$ that we observed at $T\in [2.275,2.325]$ signify that $T_c\approx 2.3$.

2.2. Astrocyte Calcium Model

We modeled astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ dynamics using the compact biophysical GCh-I astrocyte model that assumes homogeneous $\left[{\mathrm{Ca}}^{2+}\right]$ throughout the astrocyte (De Pittà, Ben-Jacob, & Berry, 2019; Matrosov et al., 2019). This agrees with experimental findings showing that $\left[{\mathrm{Ca}}^{2+}\right]$ exhibits the same response pattern to changes in neuronal activity in both astrocytic somata and microdomains (Xie et al., 2012). The GCh-I model can be generally stated as

$$\frac{d{C}_A}{dt}=F_{C_A}(C_A,I_A,h_A),$$ (2.14)

$$\frac{d{I}_A}{dt}=F_{I_A}(C_A,I_A)+J_{\beta }(C_A,\gamma ),$$ (2.15)

$$\frac{d{h}_A}{dt}=F_{h_A}(C_A,I_A,h_A),$$ (2.16)

where $\left[{\mathrm{Ca}}^{2+}\right]$ wave dynamics are modeled as a calcium-induced calcium release produced from the interaction of $\left[{\mathrm{Ca}}^{2+}\right]$ ($C_A$ in equation 2.14) with inositol 1,4,5-trisphosphate concentration, $I_A$, described by nonlinear terms $F_{C_A}$, $F_{h_A}$, and $F_{I_A}$, and the production of $I_A$ from extracellular glutamate concentration, $\gamma$, is described by nonlinear term $J_{\beta }$ (Lallouette, De Pittà, & Berry, 2019). In line with the compact modeling approach, we increased GCh-I's single input capacity to handle $n$ neuronal presynaptic inputs by expanding

$$J_{\beta }(C_A,\gamma )=O_{\beta }·{\mathcal{H}}_{0.7}\left(\gamma ,K_N\left(1+\zeta {\mathcal{H}}_1\left(C_A,K_{KC}\right)\right)\right)$$ (2.17)

into a weighted linear combination,

$$J_{\beta }(C_A,\gamma )=\sum _{i=1}^n\left[O_{{\beta }_i}·{\mathcal{H}}_{0.7}\left({\gamma }_i,K_N\left(1+\zeta {\mathcal{H}}_1\left(C_A,K_{KC}\right)\right)\right)\right],$$ (2.18)

where ${\gamma }_i$ is the extracellular glutamate concentration produced by the neuronal presynaptic input $i$ from equation 2.4 (see section 2.1), $O_{{\beta }_i}$ is the maximal conversion rate of extracellular glutamate to $I_A$ that primarily depends on glutamate receptor expression and from here on is referred to as the astrocyte tripartite weight of input $i$, ${\mathcal{H}}_u(l,v)=l^u/(l^u+v^u)$ is the Hill function with midpoint $v\in \mathbb{R}$, coefficient $u\in \mathbb{R}$, and input variable $l\in \mathbb{R}$, and all other parameters are constants taken from the GCh-I FM model in De Pittá, Goldberg, Volman, Berry, and Ben-Jacob (2009). Given 65,536 presynaptic inputs, our expanded GCh-I model is comparable to a large process or a subdomain of a whole human astrocyte that is known to integrate activity anywhere from 270,000 to 2 million synaptic inputs (Vasile et al., 2017).

2.3. Numerical Methods

We updated neuron states using a parallelized version of Monte Carlo Markov chain (MCMC) algorithm designed for GPU implementation (Preis, Virnau, Paul, & Schneider, 2009). While the standard MCMC algorithm updates one neuron state at a time, the parallelized algorithm updates neurons in batches of nonneighboring neurons. Specifically, we split the 2D lattice into two batches of nonneighboring neurons, both of which we updated using the transition probability defined by equation 2.2. In the following sections, a single iteration of neuron updates refers to the updating of all neurons following the procedure described in section 2.1. We updated neurons at the edge of the lattice using periodic boundary conditions (Yeomans, 2010).

Due to the inherent statistical randomness of updating Ising neuron states, we repeated all tests for each of the four randomly generated neuronal network models used in all experiments with 10 randomly initialized tests per model per each change in dynamics parameter $T$. This ensured that our results are not the product of a specific set of lattice couplings or neuron state initialization.

We simulated the GCh-I model using the forward Euler integration method with a time step of 1 ms.

The code is available at https://github.com/combra-lab/ICA.

3. Results

3.1. The Inverse Scaling Law

Astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ waves are known to respond in a frequency-dependent manner to neuronal activity (Schummers et al., 2008; Shao & McCarthy, 1995; Wallach et al., 2014). Indeed, using neuronal network spike activity as presynaptic input to drive the GCh-I astrocyte model (De Pittá et al., 2009) triggers $PLC\beta$ production of $I_A$, which in turn results in waves of intercellular $\left[{\mathrm{Ca}}^{2+}\right]$ (see Figures 2A and 2B) (see sections 2.1 and 2.2). To analyze the dependence of $\left[{\mathrm{Ca}}^{2+}\right]$ oscillation frequency, ${\nu }_C$, on tripartite weight and input spike frequency, we stimulated the GCh-I model for 300 seconds using a single presynaptic input that had biologically relevant spike frequencies of ${\nu }_S\in [0.02,20]$ Hz and tripartite weight ranges of $O_{\beta }\in [0,180]$$\mu \mathrm{M}$${\mathrm{s}}^{-1}$. Interpolation of ${\nu }_C$ over grid sampled parameters $(O_{\beta },{\nu }_S)$ reveals that ${\nu }_C$ experiences a sharp, nonlinear rise from 0 Hz to 0.1 Hz (see Figure 2C), which is consistent with experimentally observed frequency ranges (Timofeeva, 2019). The input ${\nu }_S$ corresponding to the middle of the ${\nu }_C$ range, ${\nu }_{S,\theta }$, depends on $O_{\beta }$. This dependence relationship resembles inverse scaling, expressed as

Figure 2:

Astrocyte calcium ion concentration wave frequency response to presynaptic spike input. (A, B) A spike at each presynaptic site releases a pulse of extracellular glutamate (see section 2.1) that stimulates astrocyte glutamate receptors, which trigger a phospholipase $C\beta$, $PLC\beta$, induced elevation of inositol 1,4,5-trisphosphate concentration, $\left[I{P}_3\right]$, which in turn produces a calcium-induced calcium release leading to intercellular calcium ion concentration, $\left[{\mathrm{Ca}}^{2+}\right]$, waves (see section 2.2). (C) Astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ wave frequency with respect to presynaptic spike input rate and maximal conversion rate of extracellular glutamate into $I{P}_3$ (astrocyte tripartite input weight). The dashed curve represents inverse scaling expressed by equation 3.1.

$${\nu }_{S,\theta }=\alpha \frac{1}{O_{\beta }},$$ (3.1)

where $\alpha$ is the scaling factor. This indicates that the responsiveness of ${\nu }_C$ to changes in ${\nu }_S$ is determined by $O_{\beta }$. Specifically, $O_{\beta }<3.0$$\mu \mathrm{M}$${\mathrm{s}}^{-1}$ produces no $\left[{\mathrm{Ca}}^{2+}\right]$ oscillations, while $O_{\beta }>60.0$$\mu \mathrm{M}$${\mathrm{s}}^{-1}$ gives rise to the full range of ${\nu }_C$, which becomes progressively steeper as it is constrained to smaller ranges of ${\nu }_S$ with larger $O_{\beta }$.

While the above analysis demonstrates the inverse relationship between input ${\nu }_S$ and $O_{\beta }$, its biologically unrealistic, single presynaptic input also imposes biologically implausible values for $O_{\beta }$, which typically range below 5 $\mu \mathrm{M}$${\mathrm{s}}^{-1}$ (De Pittá et al., 2009; Höfer, Venance, & Giaume, 2002; Stimberg, Goodman, Brette, & De Pittá, 2019). Interestingly, extending the GCh-I model to a biologically relevant number of presynaptic inputs (Vasile et al., 2017) results in biologically realistic tripartite weight values, $O_{{\beta }_i}$, for each synapse $i$ (see section 2.2). The inverse scaling relationship in Figure 2C remains provided that presynaptic input ${\nu }_S$ is replaced with the presynaptic input spike rate averaged over all inputs, ${\overline{\nu }}_S$, and $O_{\beta }$ is treated as the sum of individual tripartite weights: $O_{\beta }={\sum }_i{O}_{{\beta }_i}$. We proceeded to drive the extended GCh-I model with presynaptic input activity from an attractor neural network (see section 2.1).

3.2. Astrocyte Calcium as a Proxy of Historical Network Activity

To test astrocyte receptivity to a critical phase transition in neural activity, we looked at the dependence relationship of ${\nu }_C$ on network dynamics $T$. Given that ${\nu }_C$ significantly depends on $O_{\beta }$ (see Figure 2C) and astrocyte glutamate receptor expression undergoes bidirectional inverse scaling as a function of neuronal activity over the course of 4 to 6 hours (Xie et al., 2012), we hypothesized that astrocytes could use their glutamate receptor plasticity to learn long-term presynaptic activity patterns and thereby signal rapid deviations in network dynamics through ${\nu }_C$. In line with Xie et al. (2012), we used relation 3.1 to model glutamate receptor plasticity by initializing each $O_{{\beta }_i}$ as a function of its corresponding presynaptic input ${\nu }_{S_i}$. Specifically, we randomly initialized neuron states and updated them at $T_0$ dynamics for 10,000 iterations resulting in stabilized neuronal spike rates. Then, to obtain a representation of long-term presynaptic input activity, we further updated the network for 1200 iterations and computed the average spike rate of each input $i$, which we used to initialize each corresponding $O_{{\beta }_i}$ through equation 3.1 with $\alpha =180$. Similar to Xie et al. (2012), after initializing astrocyte tripartite weights, we tested their impact on ${\nu }_C$. First, we measured ${\nu }_{C,0}$ by further updating the neuronal network states with $T_0$ dynamics that drove the GCh-I model for 2700 iterations or 135 seconds (see section 2.1). Next, we obtained ${\nu }_{C,1}$ by changing $T_0$ to $T_1$ and further updated neuronal states for 2700 iterations (see Figure 3A). Repeating this experiment 10 times for each $(T_0,T_1)$ pair over four randomly generated networks (see section 2.1.1), we evaluated the relationship between changes in dynamics, $\Delta T=T_1-T_0$, and the corresponding changes in $\left[{\mathrm{Ca}}^{2+}\right]$ oscillation frequency, $\Delta {\nu }_C={\nu }_{C,1}-{\nu }_{C,0}$.

Figure 3:

Astrocyte calcium wave frequency response to phase change in presynaptic input activity. (A) Examples of astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ wave frequency change in response to transitions from initial near-critical presynaptic input dynamics ($T_0\approx T_c$) to (left) subcritical ($T_1<T_c$) activity and (right) supercritical ($T_1>T_c$) activity. (B) Relative change in $\left[{\mathrm{Ca}}^{2+}\right]$ wave frequency as a function of change in input dynamics from $T_0$ to $T_1$. (C) (Top) Relationship between change in $\left[{\mathrm{Ca}}^{2+}\right]$ wave frequency and corresponding change in average input spike rate with correlation for each $T_0$ group is quantified by (bottom) Pearson correlation coefficient. (D) Robustness of astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ response to inverse scaling factors $\alpha$ used to initialize tripartite weights through equation 3.1 at $T_0=2.3$. (E) The frequency of astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ waves at (left) $T_0$ dynamics and (right) resulting $T_1$ dynamics. (F) Distribution of (left) presynaptic input spike rates that result in corresponding (right) astrocyte tripartite weight distributions based on relationship 3.1 with $\alpha =180$. (B, D–F) Data points are averaged over four networks with 10 samples per network per ($T_0$, $T_1$) pair. Error bars are standard error. (B–F) See the appendix for curve fit parameters.

The $\Delta {\nu }_C$ response is positive for progressively supercritical presynaptic dynamics ($T_1>T_c$) and negative for progressively subcritical dynamics ($T_1<T_c$) (see Figure 3A). This response pattern is most pronounced when $O_{{\beta }_i}$ are initialized on near-critical ($T_0=T_c=2.3$) and subcritical ($T_0=2.2$) dynamics (see Figure 3B), which aligns with experimental evidence suggesting the brain maintains near-critical dynamics (Beggs & Plenz, 2003; Hahn et al., 2017; Shriki et al., 2013) with a tendency toward the subcritical phase (Priesemann et al., 2014). Alternatively, ${\nu }_C$ becomes unresponsive to changes in presynaptic dynamics when $O_{{\beta }_i}$ are initialized on progressively supercritical dynamics, $T_0\in \{2.4,2.5,2.6\}$. As in Figure 2C, the change in ${\nu }_C$ is nonlinear, which we approximated with a hyperbolic tangent (see the appendix). To ensure that the observed $\Delta {\nu }_C$ response is the result of changes in network dynamics and not network spike rates, we measured the correlation between $\Delta {\nu }_C$ and $\Delta {\overline{\nu }}_S={\overline{\nu }}_{S,1}-{\overline{\nu }}_{S,0}$ (see Figure 3C). Indeed, the Pearson correlation coefficient is near zero for $T_0\in \{2.2,2.3,2.4\}$ where the $\Delta {\nu }_C$ response is most pronounced, which confirms that the astrocyte response is caused by changes in network dynamics. As expected, correlation increases for $T_0\in \{2.5,2.6\}$ where both $\Delta {\nu }_C$ and $\Delta {\overline{\nu }}_S$ do not exceed $\pm 15\%$.

Although the $\Delta {\nu }_C$ response is robust to a wide range of inverse scaling factors, $\alpha$ in equation 3.1, the $\Delta {\nu }_C$ magnitude decreases for larger scaling factors (see Figure 3D). Specifically, we measured $\Delta {\nu }_C$ response for $\alpha \in \{180,250,400\}$. This sensitivity is due to the curvature of ${\nu }_C$ (see Figure 2C), since larger and smaller $\alpha$ shift the initial ${\nu }_C$ to higher and lower frequencies (see Figure 3E), respectively, where ${\nu }_C$ slopes more moderately with input spike frequency. Similar to how neurons maintain maximal dynamic range (Rasmussen, Schwartz, & Chase, 2017), this suggests that astrocytes may also have dynamic range adaptation mechanisms that maximize the responsiveness of ${\nu }_C$ to input ${\overline{\nu }}_S$. This also indicates that the astrocyte response is not purely an artifact of specific scaling factors but rather the result of differences between the two phases of network activity and their interaction with astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ dynamics.

We attributed the astrocyte's conditional response to the distinct presynaptic input spike frequency distribution of each phase and its impact on $O_{{\beta }_i}$ initialization. Specifically, supercritical dynamics $(T_0=2.6)$ exhibit an approximately gaussian ${\nu }_{S_i}$ distribution (see Figure 3F, left), which produces minimal variation in the distribution of $O_{{\beta }_i}$ during initialization (see Figure 3F, right). Transitioning to subcritical dynamics changes the spike rates to an approximately symmetrical bimodal ${\nu }_{S_i}$ distribution, keeping overall stimulation to the astrocyte approximately the same. Conversely, initializing on the subcritical $(T_0=2.2)$ bimodal ${\nu }_{S_i}$ distribution results in disproportionately larger $O_{{\beta }_i}$ for corresponding low ${\nu }_{S_i}$. Hence, subsequent changes to supercritical dynamics produce an overall increase in stimulation to the astrocyte because the increase in stimulation from inputs with large $O_{{\beta }_i}$ overcompensate for the decrease in stimulation from inputs with low $O_{{\beta }_i}$.

3.3. An Information Theoretic Model for Astrocyte Calcium Activity

The bidirectional inverse scaling of astrocyte glutamate receptor expression enables the GCh-I model to approximate and memorize the information content of long-term spike activity that is exhibited by its neuronal presynaptic inputs. The assignment of equation 3.1's disproportionately larger $O_{{\beta }_i}$ values to presynaptic inputs with lower spike frequencies than to presynaptic inputs with higher spike frequencies suggests that the GCh-I model can approximate the information-theoretic measure of self-information (surprisal), which can be defined as the information content of a spike event,

$$I_i({\sigma }_i=1)=-{log}_{10}\left(p({\sigma }_i=1)\right),$$ (3.2)

where $p({\sigma }_i=1)$ is the probability of a spike passing through synapse $i$ (Jones, 1979). To confirm this hypothesis, we resorted again to the single presynaptic input analysis, as was done for Figure 2C. Specifically, self-information of spike events can be obtained from the long-term input spike rate, ${\widehat{\nu }}_S$ (memorized by $O_{\beta }$) by restating equation 3.2 as

$$I\left({\widehat{\nu }}_S\right)=-{log}_{10}\left(p\left({\widehat{\nu }}_S\right)\right),$$ (3.3)

where probability function $p\left({\widehat{\nu }}_S\right)={\widehat{\nu }}_S/{\nu }_{S,max}$ and ${\nu }_{S,max}=20$ Hz is the maximum spike rate of a presynaptic input. To mimic the direct stimulation of the astrocyte by the current spike rate, ${\nu }_S$, we introduced the weight, $p\left({\nu }_S\right)$, to $I\left({\widehat{\nu }}_S\right)$ as follows:

$$I^{*}({\nu }_S,{\widehat{\nu }}_S)=-p\left({\nu }_S\right){log}_{10}\left(p\left({\widehat{\nu }}_S\right)\right).$$ (3.4)

This weighted self-information measure increases when memorized input spike rates are small and current input spike rates are large (see Figure 4A). Indeed, this general trend is replicated by the GCh-I model provided that the tripartite weight $O_{\beta }$ axis in Figure 2C is rescaled to the corresponding presynaptic input spike rates (see Figure 4C) using relation 3.1. Applying a hyperbolic tangent function to $I^{*}({\nu }_S,{\widehat{\nu }}_S)$ (see Figure 4B) and comparing it to the biophysical astrocyte model's ${\nu }_C$, we confirmed that both are functionally identical (see Figures 4B and 4C and the appendix). Specifically, both exhibit a step-like response when ${\widehat{\nu }}_S$ decrease and ${\nu }_S$ increase past a certain threshold. Assuming that this single input analysis extends to multiple inputs, this suggests that $I^{*}({\nu }_S,{\widehat{\nu }}_S)$ can approximate the astrocyte ${\nu }_C$ response to changes in network dynamics.

Figure 4:

Astrocyte encoding the information content of input spike activity. Comparison of biophysical astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ wave frequency response to the information content of presynaptic spike activity and the information-theoretic astrocyte (IA) model. (A) Weighted self-information of a spike event as a function of memorized presynaptic input spike rate and current input spike rate, evaluated using equation 3.4. (B) The IA model is evaluated as a hyperbolic tangent activation function applied to weighted self-information in panel A. (C) Astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ wave frequency as a function of memorized input spike rate and current input spike rate. (D) Relative change in weighted self-information as a function of change in presynaptic input dynamics, evaluated using equation 3.5. (E) Relative change in the frequency of the IA model as a function of change in presynaptic input dynamics, evaluated using equation 3.6. (D, E) Data points are averaged over four networks with 10 samples per network per ($T_0$, $T_1$) pair. Error bars are standard error. See the appendix for curve fit parameters.

Indeed, weighted self-information reproduces the biophysical astrocyte $\Delta {\nu }_C$ response to deviations in presynaptic network dynamics shown in Figure 3B. Extending equation 3.4 to multiple presynaptic inputs,

$${\overline{I}}^{*}=\frac{1}{n}\sum _{i=1}^n{I}_i^{*}({\nu }_{S_i},{\widehat{\nu }}_{S_i}),$$ (3.5)

we observed that $\Delta {\overline{I}}^{*}$ reproduces the general form of the biophysical astrocyte's $\Delta {\nu }_C$ response (see Figure 4D) on the same presynaptic spike input data that we used for the GCh-I model in Figure 3B. To achieve biologically plausible frequency output, we rescaled equation 3.5 to the same frequency range as ${\nu }_C$ by introducing an activation function. This results in the information-theoretic astrocyte (IA) model,

$${\nu }_{IA}=\varphi \left({\overline{I}}^{*}\right),$$ (3.6)

where ${\nu }_{IA}$ is the output frequency and $\varphi$ is the hyperbolic tangent activation function defined in the appendix as equation A.1 with parameters $(b_0,b_1,b_2,b_3)=(0.05,4.0,0.25,0.05)$ (see Figure 4E). This confirms that the biophysical GCh-I astrocyte model with bidirectional glutamate receptor scaling functionally approximates weighted self-information.

4. Discussion

In this letter, we propose a computational link between the well-established astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ signaling and the widely observed self-organized critical dynamics of the brain. Our results suggest that astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ waves signal network deviations away from critical dynamics by integrating current (on order of seconds) neuronal presynaptic activity conditioned on a learned memory of longer-term (on the order of many hours) activity. We show that this function arises from the bidirectional scaling of astrocyte glutamate receptor expression, which allows astrocytes to approximate and memorize the information content of their presynaptic inputs. This led to a computationally simple, information-theoretic model that functionally approximated the biophysical astrocyte model. We confirmed the robustness of our results through randomized testing.

Our computational results (see Figure 3) can suggest several hypotheses that can be experimentally tested, including one where deviations away from critical neuronal network dynamics are signaled by corresponding changes in the frequency of astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ waves. Interestingly, the functional significance of $\left[{\mathrm{Ca}}^{2+}\right]$ signaling is attributed to the astrocytic release of gliotransmitters, such as glutamate, into the synaptic extracellular space, which are well known to regulate synaptic plasticity at multiple spatiotemporal scales (De Pittà et al., 2016). Additionally, computational studies have suggested how both feedback and feedforward astrocyte gliotransmission mechanisms can tune the balance between potentiation and depression in the plasticity of single synapses (De Pittà & Brunel, 2016). Since synaptic plasticity requires a fine-tuned balance between potentiation and depression to maintain near-critical network dynamics (Ivanov & Michmizos, 2021; Stepp, Plenz, & Srinivasa, 2015), our results put forth an exciting possibility that $\left[{\mathrm{Ca}}^{2+}\right]$-dependent gliotransmission mechanisms are involved in the brain's self-organization near the critical phase transition. Indeed, astrocyte-like spatiotemporal plasticity modulation can stabilize the dynamics of brain-inspired neuronal networks near criticality, thereby improving their learning performance (Ivanov & Michmizos, 2021). This is further supported by experimental evidence showing that astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ activity plays a causal role in neuronal network state switching (Poskanzer & Yuste, 2011, 2016), which is linked to critical brain dynamics and cognitive states (Chialvo, 2010; Kelso et al., 1992).

The continuous nature of the $\left[{\mathrm{Ca}}^{2+}\right]$ wave frequency response suggests astrocytes not only encode the direction of deviations in dynamics (see Figures 2C and 3B), but also their distance from the critical transition. Interestingly, specific brain dynamics located on a continuum around the critical phase transition (Gautam, Hoang, McClanahan, Grady, & Shew, 2015; Hahn et al., 2010, 2017; Pasquale, Massobrio, Bologna, Chiappalone, & Martinoia, 2008; Tetzlaff, Okujeni, Egert, Wörgötter, & Butz, 2010) correspond to particular cognitive states (Meisel et al., 2017). For instance, critical dynamics are most pronounced in conscious restful wakefulness (Bellay et al., 2015; Priesemann et al., 2013; Tagliazucchi et al., 2016); slightly subcritical dynamics are associated with states of rapid eye movement sleep (Priesemann et al., 2013) and attention (Fagerholm et al., 2015), and marginally supercritical dynamics are linked to slow wave sleep (Priesemann et al., 2013) and anaesthesia (Bellay et al., 2015). Given that these and other cognitive states are well known to be regulated by astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ signaling (Bojarskaite et al., 2020; Foley et al., 2017; Halassa & Haydon, 2010; Ingiosi et al., 2020; Thrane et al., 2012), our results suggest critical dynamics as the mechanistic link between astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ signaling and cognitive state regulation.

By establishing a link between critical brain dynamics and astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ signaling (see Figure 3), our results also suggest that astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ signaling plays an important role in the brain's ability to process information, since the capacity to process information is maximized in critical systems through the balancing of two opposing computational processes: learning and memory (Bertschinger & Natschläger, 2004; Legenstein & Maass, 2007). Indeed, both processes have long been associated with $\left[{\mathrm{Ca}}^{2+}\right]$ wave signaling in astrocytes (Perea & Araque, 2010; Poskanzer & Yuste, 2016). In fact, the human brain's unprecedented function has been linked to the evolutionary development of its astrocytes, namely, through an increased number of tripartite connections and more refined $\left[{\mathrm{Ca}}^{2+}\right]$ signaling (Oberheim et al., 2006, 2009; Zhang, Sloan et al., 2016). Both attributes are crucial for the proper operation of our proposed astrocyte function. This suggests that the evolutionary enhancement of either component may have contributed to better monitoring of network dynamics, which in turn contributed to the increased human functional competence. Interestingly, learning and memory improved in rodents that received human progenitor cells that produced human-like astrocytes (Han et al., 2013).

The ability to detect changes in network dynamics as a result of receptor plasticity (see Figure 3) suggests that astrocyte learning mechanisms can significantly extend the information processing capacity of astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ signaling beyond integration of neuronal inputs (Gordleeva, Ermolaeva, Kastalskiy, & Kazantsev, 2019). Indeed, it is well known that synaptic plasticity can enable neurons to carry out complex functions such as extraction of principal components (PCA; Gilson, Fukai, & Burkitt, 2012) or approximation of the expectation-maximization (EM) algorithm (Nessler, Pfeiffer, & Maass, 2009). Similarly, our results (see Figure 4) put forth an experimentally testable hypothesis where bidirectional scaling of glutamate receptors (Xie et al., 2012) facilitates astrocytes to approximate and memorize the information content of individual presynaptic inputs. This hypothesis is supported by experimental evidence suggesting astrocytes use neuronal activity to perform additional computations to modulate neuronal networks (Gómez-Gonzalo et al., 2014; Navarrete et al., 2012; Papouin, Dunphy, Tolman, Dineley, & Haydon, 2017) or extend the information processing performed by neurons (Schummers et al., 2008).

The functional equivalence of the biophysical astrocyte model to a computationally simple information-theoretic model (see Figure 4) facilitates computational investigations of emergent network behavior in large-scale neuron-astrocyte networks. Indeed, astrocytes are well known to form extensive networks (Giaume, Koulakoff, Roux, Holcman, & Rouach, 2010) that primarily use $\left[{\mathrm{Ca}}^{2+}\right]$ signaling as a means of communication in between both astrocytes, and astrocytes and neurons (Dallérac & Rouach, 2016; Halassa & Haydon, 2010; Perea, Sur et al., 2014). Yet investigation of the functional roles of astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ signaling in network-level functions such as learning, memory, and network dynamics is currently limited by the lack of biologically plausible and computationally simple astrocyte models that can scale to large neuron-astrocyte networks. Similar to how simple neuron models enabled widespread computational study of emergent network behavior in large neuronal networks (Herz, Gollisch, Machens, & Jaeger, 2006; Izhikevich, 2003), our proposed information-theoretic astrocyte model can facilitate the exploration of network-level phenomena in large-scale neuron-astrocyte networks (Goldberg, De Pittà, Volman, Berry, & Ben-Jacob, 2010).

Here, we modeled a single astrocyte compartment with $\left[{\mathrm{Ca}}^{2+}\right]$ dynamics externally driven by multiple presynaptic inputs that were integrated as a weighted sum. This assumed that $\left[{\mathrm{Ca}}^{2+}\right]$ was the same for all presynaptic inputs. Since the number of synaptic inputs constrained the interpretation of our compartment to a single process, reducing the number of synaptic inputs would facilitate interpretation to smaller compartments such as finer processes or microdomains. Hence, the weighted-sum approach for integrating multiple sets of synaptic inputs is expandable to multicompartment astrocyte models with compartments varying in size and $\left[{\mathrm{Ca}}^{2+}\right]$ dynamics (Matrosov et al., 2019). Such multicompartment models are important for investigation of the observed heterogeneity in $\left[{\mathrm{Ca}}^{2+}\right]$ dynamics across spatially localized astrocytic domains (Stobart et al., 2016).

Overall, our study links several areas of neuroscience with computational modeling to provide insight into the functional role of astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ waves in brain information processing. While not a substitute for experimental findings, computational models have long been successful in generating interest in new hypotheses, some of which were eventually evaluated experimentally. Similarly, we hope that the modeling results we presented here will drive several testable hypotheses, elucidating the roles of astrocyte $\left[{\mathrm{Ca}}^{2+}\right]$ signaling in the brain's function and dysfunction.

Appendix

Function Fitting and Parameters

We fitted hyperbolic tangent curves of the form

$$y=b_{0}tanh\left(b_1\left(x-b_2\right)\right)+b_3,$$ (A.1)

All parameters are in Table 2 with figure and plot-specific references shown.

Table 2:

Hyperbolic Tangent Curve Fit Parameters.

Figure	${\mathbf{b}}_{\mathbf{0}}$	${\mathbf{b}}_{\mathbf{1}}$	${\mathbf{b}}_{\mathbf{2}}$	${\mathbf{b}}_{\mathbf{3}}$	${\mathbf{R}}^{\mathbf{2}}$
Figure 3B (purple)	65.810	3.477	0.130	38.112	0.981
Figure 3B (blue)	48.869	3.742	0.051	21.332	0.987
Figure 3B (green)	24.530	3.065	−0.055	9.451	0.940
Figure 3B (yellow)	6.827	4.509	−0.113	4.132	0.870
Figure 3B (orange)	3.757	12.00	−0.297	−2.187	0.820
Figure 3D ($\alpha =400$)	0.013	3.641	0.051	0.065	0.985
Figure 3D ($\alpha =250$)	0.015	3.724	0.062	0.053	0.983
Figure 3D ($\alpha =180$)	0.017	3.874	0.055	0.043	0.987
Figure 3E (right, $\alpha =400$)	21.144	3.570	0.0522	8.937	0.983
Figure 3E (right, $\alpha =250$)	33.423	3.731	0.060	15.522	0.982
Figure 3E (right, $\alpha =180$)	48.869	−3.742	0.051	21.332	0.987
Figure 4D (purple)	90.610	4.886	0.206	72.991	0.988
Figure 4D (blue)	52.315	5.426	0.115	36.219	0.987
Figure 4D (green)	21.441	5.508	0.031	13.490	0.943
Figure 4D (yellow)	3.883	18.089	−0.010	4.300	0.883
Figure 4D (orange)	2.248	8.346	−0.141	−1.191	0.762
Figure 4E (purple)	57.625	5.177	0.141	41.046	0.991
Figure 4E (blue)	37.846	5.095	0.071	21.466	0.989
Figure 4E (green)	16.814	4.758	−0.008	7.654	0.946
Figure 4E (yellow)	4.509	4.294	−0.068	2.140	0.886
Figure 4E (orange)	2.664	7.997	−0.176	−1.781	0.858

We fitted polynomials of the form:

$$\begin{array}{ccc}\hfill y& =& b_{10}{x}^{10}+b_{11}{x}^9+b_{12}{x}^8+b_{13}{x}^7+b_{14}{x}^6+b_{15}{x}^5+b_{16}{x}^4+b_{17}{x}^3\hfill \\ & & +b_{18}{x}^2+b_{19}x+b_{20}\hfill \end{array}$$ (A.2)

with degrees including 0, 1, 5, and 10. All parameters are in Table 3 with figure and plot-specific references shown.

Table 3:

Polynomial Curve Fit Parameters.

Figure	Degree	${\mathbf{b}}_{\mathbf{10}}$	${\mathbf{b}}_{\mathbf{11}}$	${\mathbf{b}}_{\mathbf{12}}$	${\mathbf{b}}_{\mathbf{13}}$	${\mathbf{b}}_{\mathbf{14}}$	${\mathbf{b}}_{\mathbf{15}}$	${\mathbf{b}}_{\mathbf{16}}$	${\mathbf{b}}_{\mathbf{17}}$	${\mathbf{b}}_{\mathbf{18}}$	${\mathbf{b}}_{\mathbf{19}}$	${\mathbf{b}}_{\mathbf{20}}$	${\mathbf{R}}^{\mathbf{2}}$
Figure 3C (top, purple)	1	1.77	23.85										0.019
Figure 3C (top, blue)	1	−0.33	16.94										0.001
Figure 3C (top, green)	1	0.14	10.57										0.001
Figure 3C (top, yellow)	1	1.39	4.68										0.239
Figure 3C (top, orange)	1	1.65	0.83										0.539
Figure 3C (bottom)	2	8.77	−40.36	46.47									0.934
Figure 3E (left, $\alpha =400$)	0	0.056											–
Figure 3E (left, $\alpha =250$)	0	0.045											–
Figure 3E (left, $\alpha =180$)	0	0.035											–
Figure 3F (left, purple)	10	−4.86 × 10−9	4.85 × 10−7	−2.07 × 10−5	4.91 × 10−4	−7.12 × 10−3	6.49 × 10−2	−0.37	1.29	−2.51	2.13	−1.46	0.979
Figure 3F (left, blue)	10	−6.50 × 10−9	6.51 × 10−7	−2.78 × 10−5	6.64 × 10−4	−9.67 × 10−3	8.88 × 10−2	−0.51	1.81	−3.63	3.47	−2.21	0.933
Figure 3F (left, green)	10	−7.75 × 10−9	7.76 × 10−7	−3.32 × 10−5	7.92 × 10−4	−1.16 × 10−2	1.07 × 10−1	−0.62	2.20	−4.52	4.66	−3.12	0.934
Figure 3F (left, yellow)	10	−8.57 × 10−9	8.58 × 10−7	−3.67 × 10−5	8.76 × 10−4	−1.28 × 10−2	1.18 × 10−1	−0.69	2.46	−5.16	5.69	−4.31	0.986
Figure 3F (left, orange)	10	−1.02 × 10−8	1.03 × 10−6	−4.38 × 10−5	1.05 × 10−3	−1.53 × 10−2	1.41 × 10−1	−0.82	2.98	−6.38	7.53	−6.34	0.995
Figure 3F (right, purple)	5	−3.76 × 1010	2.29 × 109	−4.49 × 107	3.65 × 105	−1.40 × 103	4.58 × 101						0.987
Figure 3F (right, blue)	5	−6.33 × 1010	3.17 × 109	−5.50 × 107	4.08 × 105	−1.54 × 103	5.69 × 101						0.994
Figure 3F (right, green)	5	−1.05 × 1011	4.52 × 109	−7.05 × 107	4.81 × 105	−1.82 × 103	7.42 × 101						0.997
Figure 3F (right, yellow)	5	−2.25 × 1011	1.10 × 1010	−1.64 × 108	9.98 × 105	−3.17 × 103	1.52						0.997
Figure 3F (right, orange)	5	−1.26 × 1013	2.33 × 1011	−1.66 × 109	5.57 × 106	−9.53 × 103	3.89						0.999