We begin by defining the metrics used to quantify avalanche statistics and briefly summarize experimental observations, which have been reviewed in detail elsewhere (Plenz et al., 2021; O’Byrne and Jerbi, 2022; Girardi-Schappo, 2021). Activity is recorded across a set of neurons and binned in time. Avalanches are then defined as contiguous time bins, in which at least one neuron in the population is active. The duration of an avalanche is the number of contiguous time bins and the size is the summed activity during the avalanche. The distributions of avalanche size and duration are fit to power laws (${\displaystyle P(S)\sim S^{-\tau }}$ for size ${\displaystyle S}$, and ${\displaystyle P(D)\sim D^{-\alpha }}$ for duration ${\displaystyle D}$) using standard methods (Clauset et al., 2009).

Power laws can be indicative of criticality, but they can also result from non-critical mechanisms (Touboul and Destexhe, 2017; Priesemann and Shriki, 2018). A more stringent test of criticality is the ‘crackling’ relationship (Perkovic et al., 1995; Touboul and Destexhe, 2017), which involves fitting a third power-law relationship, ${\displaystyle \overline{S}(D)\sim D^{{\gamma }_{\text{fit}}}}$, and comparing ${\displaystyle {\gamma }_{\text{fit}}}$ to the predicted exponent ${\displaystyle {\gamma }_{\text{pred}}}$, derived from the size and duration exponents, ${\displaystyle \tau }$ and ${\displaystyle \alpha }$:

Previous work demonstrating approximate power laws in size and duration distributions through the mechanism of a slowly changing latent variable did not generate crackling (Touboul and Destexhe, 2017; Priesemann and Shriki, 2018).

Measuring power-laws in empirical data is challenging: it generally requires setting a lower cut-off in the size and duration, and the power-law behavior only has limited range due to the finite size and duration of the recording itself. Nonetheless, there is some consensus (Shew et al., 2015; Fontenele et al., 2019; Ma et al., 2019) that even if ${\displaystyle \tau }$ and ${\displaystyle \alpha }$ vary over a wide range (1.5 to about 3) across recordings, the values of ${\displaystyle {\gamma }_{\text{fit}}}$ and ${\displaystyle {\gamma }_{\text{pred}}}$ stay in a relatively narrow range, from about 1.1 to 1.3.

We study a model of a population of neurons that are not coupled to each other directly but are driven by a small number of latent dynamical variables – that is, slowly changing inputs that are not themselves measured (Figure 1A). We are agnostic as to the origin of these inputs: they may be externally driven from other brain areas, or they may arise from large fluctuations in local recurrent dynamics. The model was chosen for its simplicity, and because we have previously shown that this model with at least about five latent variables can produce power laws under the coarse-graining analysis (Morrell et al., 2021). In this paper, we examine avalanche criticality in the same model.

Figure 1

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Specifically, we model the neurons as binary units (${\displaystyle s_i}$) that are randomly (${\displaystyle J_{i\mu }\sim N(0,1)}$) coupled to dynamical variables ${\displaystyle h_{\mu }(t)}$. The probability of any pattern ${\displaystyle \{s_i\}}$, given the current state of the latent variables, is

where the parameter ${\displaystyle \eta }$ controls the scaling of the variables and ${\displaystyle ϵ}$ controls the overall activity level. We modeled each latent variable as an Ornstein-Uhlenbeck process with the time scale ${\displaystyle {\tau }_F}$ (see Materials and methods). Thus our model has four parameters: ${\displaystyle \eta }$ (input scaling), ${\displaystyle ϵ}$ (activity threshold), ${\displaystyle {\tau }_F}$ (dynamical timescale), and ${\displaystyle N_F}$ (number of latent variables).

Distributions of avalanche size and avalanche duration within this model followed approximate power laws (Figure 1C; see Materials and methods). In the example shown (${\displaystyle N_F=5}$, ${\displaystyle {\tau }_F={10}^4}$, ${\displaystyle \eta =4}$ and ${\displaystyle ϵ=12}$), we found exponents ${\displaystyle \tau =1.89\pm 0.02}$ (size) and ${\displaystyle \alpha =2.11\pm 0.02}$ (duration). Further, the average size of avalanches with fixed duration scaled as ${\displaystyle S\sim D^{\gamma }}$, with the fitted ${\displaystyle {\gamma }_{\text{fit}}=1.24\pm 0.02}$, in agreement with the predicted value ${\displaystyle {\gamma }_{\text{pred}}=1.24\pm 0.02}$. Thus, our model could generate avalanche scaling, at least for some parameter choices. In the following sections, we examine how avalanche scaling depends on model parameters (${\displaystyle N_F}$, ${\displaystyle {\tau }_F}$, ${\displaystyle \eta }$ and ${\displaystyle ϵ}$; see Table 2). We first focus on two sets of simulations: one set with ${\displaystyle N_F=1}$ latent variable, which does not generate scaling under coarse-graining (Morrell et al., 2021), and one set with ${\displaystyle N_F=5}$ latent variables, which can generate such scaling for some values of parameters ${\displaystyle {\tau }_F}$, ${\displaystyle \eta }$, and ${\displaystyle ϵ}$ (Morrell et al., 2021; Table 1).

We analyzed avalanches from one- and five-variable simulations, each with fixed latent dynamical timescale (${\displaystyle {\tau }_F=5\times {10}^3}$ time steps; see Table 2 for parameters). In the following sections, time is measured in simulation time steps, see Materials and methods for converting time steps to seconds. We used established methods for measuring empirical power laws (Clauset et al., 2009). The minimum cutoffs for size (${\displaystyle S_{\text{min}}}$) and duration (${\displaystyle D_{\text{min}}}$) are indicated by vertical lines in Figure 2. For the population coupled to a single latent variable, the avalanche size distribution was not well fit by a power law (Figure 2A). With a sufficiently high minimum cut-off (${\displaystyle D_{\text{min}}}$), the duration distribution was approximately power-law (Figure 2B).

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We next assessed whether the simulation produced crackling. If so, the value ${\displaystyle {\gamma }_{\text{fit}}}$ obtained by fitting ${\displaystyle \overline{S}(D)\sim D^{{\gamma }_{\text{fit}}}}$ would be similar to ${\displaystyle {\gamma }_{\text{pred}}=\frac{\alpha -1}{\tau -1}}$. In many cases, such as the one-variable example shown in Figure 2C, the full range of avalanche durations were not fit by a single power law. Therefore, we determined the largest range, over which a power law was a good fit to the simulated observations. In this case, slightly over two decades of apparent scaling were observed starting from avalanches with minimum duration slightly less than 100 time steps (Figure 2C), with a best-fit value of ${\displaystyle {\gamma }_{fit}\in [1.69,1.74]}$. As we did not find a power-law in the size distribution, calculating ${\displaystyle {\gamma }_{\text{pred}}}$ is meaningless. If we do it anyway, we obtain ${\displaystyle {\gamma }_{pred}=0.83\pm 0.03}$ (yellow line in Figure 2C), which clearly deviates from the fitted value of ${\displaystyle \gamma }$. Thus, for the single latent dynamical variable model (${\displaystyle {\tau }_F=5000}$), power-law fits are poor, and there is no crackling.

The five-variable model produces a different picture. We now find avalanches, for which size and duration distributions are much better fit by power-law models starting from very low minimum cutoffs (Figure 2D–E, Figure 2—figure supplement 2). Average size scaled with duration, again over more than two decades, with ${\displaystyle {\gamma }_{\mathrm{f}\mathrm{i}\mathrm{t}}=1.27\pm 0.03}$, which was in close agreement with ${\displaystyle {\gamma }_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}=1.25\pm 0.02}$ (Figure 2F). Holding other parameters constant, we thus found that scaling relationships and crackling arise in the multi-variable model but not the single-variable model.

Based on simulations in the previous section, we surmised that the five-variable simulation generated scaling more readily due to creating an ‘effective’ latent variable that had slower dynamics than any individual latent variable. We reasoned that at any moment in time, the latent variable state ${\displaystyle h_{\mu }(t)}$ is a vector in the latent space. Because coupling to the latent variables is random throughout the population, only the length (${\displaystyle \sim \sqrt{N_F}}$) and not the direction of this vector matters, and the timescale of changes in this length would be much slower than ${\displaystyle {\tau }_F}$, the timescale of each of the components ${\displaystyle h_{\mu }(t)}$. We therefore speculated that increasing the timescale of dynamics of the latent variables should eventually lead to scaling and crackling in the single-variable model as well as the five-variable one. To examine the dependence of avalanche scaling on this timescale, we simulated one-variable and five-variable networks at fixed ${\displaystyle \eta }$ and ${\displaystyle ϵ}$ coupled to latent variables with the correlation time of their Ornstein-Uhlenbeck dynamics of ${\displaystyle {\tau }_F\in [{10}^3,{10}^5]}$ time steps, spanning from a factor of 10 faster to a factor of 10 slower than the original ${\displaystyle {\tau }_F}$ in Figure 1. Simulations were replicated five times at each combination of parameters by drawing new latent variable coupling values (${\displaystyle J_{i\mu }}$), as well as new latent variable dynamics and instances of neural firing. For simulations that passed the criteria to be fitted by power laws, we plot the fitted values of ${\displaystyle \tau }$ , ${\displaystyle \alpha }$, ${\displaystyle {\gamma }_{\text{fit}}}$, and ${\displaystyle {\gamma }_{\text{fit}}-{\gamma }_{\text{pred}}}$ (Figure 2G–J). Missing points are those for which distributions did not pass the power law fit criteria.

In the single-variable model, best-fit exponents changed abruptly for latent variable timescale around ${\displaystyle {\tau }_F={10}^4}$ (Figure 2G and H), while in the five-variable model, exponents tended to increase gradually (Figure 2G and H, red). The discontinuity in the single-variable case reflected a change in the lower cutoff values in the power-law fits: size and duration distributions generated with faster latent dynamics could be fit reasonably well to a power law by using a high value of the lower cutoff (Figure 2—figure supplement 3). For time scales greater than ∼10⁴, the minimum cutoffs dropped, and the single-variable model generated power-law distributed avalanches and crackling (Figure 2J), similar to the five-variable model. In summary, in the latent dynamical variable model, introducing multiple variables generated scaling at faster timescales. However, by slowing the timescale of the latent dynamics, the model generated signatures of critical avalanche scaling for both multi- and single-variable simulations.

In the previous section, we found that a very slow single latent dynamical variable generated avalanche criticality in the simulation population. Thus, from now on, we simplify the model in order to characterize avalanche statistics across values of input scaling ${\displaystyle \eta }$ and firing threshold ${\displaystyle ϵ}$. Specifically, we modeled a population of ${\displaystyle N=128}$ neurons coupled to a single quasi-static latent variable. We simulated 10³ segments of 10⁴ steps each and drew a new value of the latent variable (${\displaystyle h\sim N(0,1)}$) for each segment. Ten replicates of the simulation were generated at each of the combinations of ${\displaystyle \eta }$ and ${\displaystyle ϵ}$ (see Materials and methods).

Almost independent of ${\displaystyle \eta }$ and ${\displaystyle ϵ}$, we found quality power law fits and crackling. Figure 3 shows the average (across ${\displaystyle n=10}$ network realizations) of the exponents extracted from size (${\displaystyle \tau }$, Figure 3A) and duration (${\displaystyle \alpha }$, Figure 3C) distributions. At small firing threshold (${\displaystyle ϵ=2}$), we do not observe scaling because the system is always active, and all avalanches merge into one. At large firing threshold ${\displaystyle ϵ}$ and low input scaling ${\displaystyle \eta }$, we do not observe scaling because activity is so sparse that all avalanches are small. At intermediate values of the parameters, the simulations generated plausible scaling relationships in size and duration. The difference between ${\displaystyle {\gamma }_{\text{fit}}}$ and ${\displaystyle {\gamma }_{\text{pred}}}$ was typically less than 0.1 (Figure 4D–F), which was consistent with previously reported differences between fit and predicted exponents (Ma et al., 2019). Thus, there appears to be no need for fine-tuning to generate apparent scaling in this model, at least in the limit of (near) infinite observation time. Wherever ${\displaystyle \eta }$ and ${\displaystyle ϵ}$ generate avalanches, there are approximate power-law distributions and crackling.

Figure 3

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To determine where avalanches occur, we derive the avalanche rate across values of the latent variable ${\displaystyle h}$. In the quasi-static model, the probability of an avalanche initiation is the probability of a transition from the quiet to an active state. Because all neurons are conditionally independent, this is ${\displaystyle P_{\text{ava}}=P_{\text{silence}}(1-P_{\text{silence}})}$. Then the expected number of avalanches ${\displaystyle {\hat{N}}_{\text{ava}}}$ is obtained by integrating ${\displaystyle P_{\text{ava}}}$ over ${\displaystyle h}$ at each value of ${\displaystyle \eta }$ and ${\displaystyle ϵ}$:

where ${\displaystyle p(h)}$ is the standard normal distribution. This probability tracks the observed number of avalanches across simulations, Figure 4A.

To gain an intuition for the conditions under which avalanches occur, we show two slices of the avalanche probability, at fixed ${\displaystyle \eta }$ (Figure 4B) and at fixed ${\displaystyle ϵ}$ (Figure 4C). Black regions indicate where avalanches are likely to occur. If, for a given value of ${\displaystyle ϵ}$ and ${\displaystyle \eta }$, there is no overlap between high avalanche probability regions and the distribution of ${\displaystyle h}$, then there will be no avalanches. For large ${\displaystyle ϵ}$, avalanches occur because neurons with large coupling to the latent variable (${\displaystyle \eta |J_i|>>1}$, recall ${\displaystyle J_i\sim N(0,1)}$) are occasionally activated by a value of the latent variable ${\displaystyle h}$ that is sufficient to exceed ${\displaystyle ϵ}$ (Figure 4B). Thus, the scaling parameter ${\displaystyle \eta }$ controls the value of ${\displaystyle h}$ for which avalanches occur most frequently (Figure 4C). As ${\displaystyle ϵ}$ decreases, avalanches occur for smaller and smaller ${\displaystyle h}$ until avalanches primarily occur when ${\displaystyle h=0}$.

To calculate the probability of avalanches, we must integrate over all values of ${\displaystyle h}$, but we can gain a qualitative understanding of which avalanche regime the system is in by examining the probability of avalanches at ${\displaystyle h=0}$. At ${\displaystyle h=0}$, the avalanche probability (see Materials and methods) is

which is maximized at ${\displaystyle {ϵ}_0=-\log (2^{1/N}-1)}$, independent of ${\displaystyle J_i}$ and ${\displaystyle \eta }$. After some algebra, we find that ${\displaystyle {ϵ}_0\sim \log N}$ for large ${\displaystyle N}$. The dependence on ${\displaystyle N}$ reflects that a larger threshold is required for larger networks: large networks (${\displaystyle N\to \mathrm{\infty }}$) are unlikely to achieve complete network silence, therefore preventing avalanches from occurring. Similarly, small networks (${\displaystyle N\sim 1}$) are unlikely to fire consecutively and thus are unlikely to avalanche.

We plot ${\displaystyle P_{\text{ava}}(ϵ,\eta ;J_i,N,h=0)}$ as a function of ${\displaystyle ϵ}$ in Figure 4D. The peak at ${\displaystyle {ϵ}_0}$ divides the space into two regions. For ${\displaystyle ϵ<{ϵ}_0}$, a power-law is only observed in the large-size avalanches, which are rare (Figure 4E, green). By contrast, when ${\displaystyle ϵ>{ϵ}_0}$, minimum size cutoffs are low (Figure 4F, orange). Both regions, ${\displaystyle ϵ<{ϵ}_0}$ and ${\displaystyle ϵ>{ϵ}_0}$, exhibit crackling noise scaling. If observation times are not sufficiently long (estimated in Figure 4—figure supplement 1), then scaling will not be observed in the ${\displaystyle ϵ<{ϵ}_0}$ region, whose scaling relations arise from rare events. Insufficient observation times may explain experiments and simulations where avalanche scaling was not found.

Our analysis of ${\displaystyle P_{\text{ava}}(ϵ,\eta ,h)}$ at ${\displaystyle h=0}$ suggested that there are two types of avalanche regimes: one with high activity and high minimum cutoffs in the power law fit (Type 1), and the other with lower activity and size cutoffs (Type 2). Further, when ${\displaystyle P_{\text{ava}}}$ drops to zero, avalanches disappear because the activity is too high or too low. We now examine how information about the value of the latent variables represented in the network activity relates to the activity type. To delineate these types, we calculated numerically ${\displaystyle {ϵ}^{\ast }(\eta )}$, the value of ${\displaystyle ϵ}$, for which the probability of avalanches is maximized, and the contours of ${\displaystyle P_{\text{ava}}}$ (Figure 5A). Curves for ${\displaystyle {ϵ}^{\ast }(\eta )}$ and ${\displaystyle {ϵ}_0}$ and ${\displaystyle P_{\text{ava}}={10}^{-3}}$ are shown in Figure 5A and B.

Figure 5

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We expect that the more cells fire, the more information they would convey, until the firing rate saturates, and inferring the value of the latent variable becomes impossible. Figure 5B supports the prediction: generally, information is higher in regions with more activity (lower ${\displaystyle ϵ}$, higher ${\displaystyle \eta }$), but only up to a limit: as ${\displaystyle ϵ\to 0}$, information decreases. This decrease begins approximately where the probability of avalanches drops to nearly zero (dashed black lines, Figure 5B–E) because all of the activity merges into a few very large avalanches. In other words, the Type-1 avalanche region coincides with the highest information about the latent variable.

The critical brain hypothesis suggests that the brain operates in a critical state, and its functional role may be in optimizing information processing (Beggs, 2008; Chialvo, 2010). Under this hypothesis, we would expect the information conveyed by the network to be maximized in the regions we observe avalanche criticality. However, we see that critical regions do not always have optimal information transmission. In Figure 5, the region that displays crackling noise is that where avalanches exist (${\displaystyle P_{\text{ava}}>0.001}$), which corresponds to any ${\displaystyle \eta }$ value and ${\displaystyle ϵ\gtrsim 3}$. This avalanche region encompasses both networks with high information transmission and networks with low information transmission. In summary, observing avalanche criticality in a system does not imply a high-information processing network state. However, the scaling can be seen at smaller cutoffs, and hence with shorter recordings, in the high-information state. This parallels the discussion by Schwab et al., 2014, who noticed that the Zipf’s law always emerges in neural populations driven by quasi-stationary latent fields, but it emerges at smaller system sizes when the information about the latent variable is high.

A wide range of critical exponents has been found in ex vivo and in vivo recordings from various systems. For instance, the seminal work on avalanche statistics in cultured neuronal networks by Beggs and Plenz, 2003 found size and duration exponents of 1.5 and 2.0 respectively, along with ${\displaystyle \gamma =2}$, when time was discretized with a time bin equal to the average inter-event interval in the system. A subset of the in vivo recordings analyzed from anesthetized cat (Hahn et al., 2010) and macaque monkeys (Petermann et al., 2009) also exhibited a size distribution exponent close to 1.5. By contrast, a survey of many in vivo and ex vivo recordings found power-law size distributions with exponents ranging from 1 to 3 depending on the system (Fontenele et al., 2019). Separately, Ma et al., 2019 reported recordings in freely moving rats with size exponents ranging from 1.5 to 2.7. In these recordings, when the crackling relationship held, the reported value of ${\displaystyle \gamma }$ was near 1.2 (Fontenele et al., 2019; Ma et al., 2019).

A model for generating avalanche criticality is a critical branching process (Beggs and Plenz, 2003), which predicts size and duration exponents of 1.5 and 2 and scaling exponent ${\displaystyle \gamma }$ of 2. However, there are alternatives: Lombardi et al., 2023 showed that avalanche criticality may also be produced by an adaptive Ising model in the sub-critical regime, and in this case, the scaling exponent ${\displaystyle \gamma }$ was not 2 but close to 1.6. Our model, across the parameters we tested that produced exponents consistent with the scaling relationship, generated ${\displaystyle \tau }$ values that ranged from 1.9 to about 2.5. Across those simulations, we found values ${\displaystyle \gamma }$ within a narrow band from 1.1 to 1.3 (see Figure 2I and J and Figure 3H). While the exponent values our model produces are inconsistent with a critical branching process (${\displaystyle \gamma =2}$), they match the ranges of exponents estimated from experiments and reported by Fontenele et al., 2019. In this context, it might be useful to explore if our model and that of Lombardi et al., 2023 might be related, with the adaptive feedback signal of the latter viewed as an effective, latent variable of the former.

A genuine challenge in comparing exponents estimated from different experiments with different recording modalities (spiking activity, calcium imaging, LFP, EEG, or MEG) arises from differences in spatial and temporal scale specific to a particular recording, as well as the myriad decisions made in avalanche analysis, such as defining thresholds or binning in time. Thus, one possible reason for differences in exponents across studies may derive from how the system is sub-sampled in space or coarse-grained in time, both of which systematically change exponents ${\displaystyle \tau }$ and ${\displaystyle \alpha }$ (Beggs and Plenz, 2003; Shew et al., 2015) and could account for differences in ${\displaystyle \gamma }$ (Capek et al., 2023). The model we presented here could be used as a test bed for examining how specific analysis choices affect exponents estimated from recordings.

A second possible explanation for differences in exponents is that different experiments study similar, but distinct biological phenomena. For instance, networks that were cultured in vitro may differ from those that were not, whether they are in vivo or ex vivo (i.e. brain slices), and sensory-processing networks may have different dynamics from networks with different processing demands. It is possible that certain networks develop connections between neurons such that they truly do produce dynamics that approximate a critical branching process, while other networks have different structure and resulting dynamics and thus can be better understood as coupled neurons receiving feedback (Lombardi et al., 2023) or as a system coupled to latent dynamical variables. This is especially true in sensory systems, where coupling to (latent) external stimuli in a way that the neural activity can be used to infer the stimuli is the reason for the networks’ existence (Schwab et al., 2014).

Our model is not the first to produce approximate power-law size and duration distributions for avalanches from a latent variable process (Touboul and Destexhe, 2017; Priesemann and Shriki, 2018). In particular, Priesemann and Shriki, 2018 derived the conditions, under which an inhomogeneous Poisson process could produce such approximate scaling. The basic idea is to generate a weighted sum of exponentially distributed event sizes, each of which are generated from a homogeneous Poisson process. How each process is weighted in this sum determines the approximate power-law exponent, allowing one to tune the system to obtain the critical values of 1.5 and 2. Interestingly, this model did not generate non-trivial scaling of size with duration (${\displaystyle S\sim D^{\gamma }}$). Instead, they found ${\displaystyle \gamma =1}$, not the predicted ${\displaystyle \gamma =2}$. Our results differ significantly, in that ${\displaystyle \gamma }$ was typically between 1.1 and 1.3 and it was nearly always close to the prediction from ${\displaystyle \alpha }$ and ${\displaystyle \tau }$. We speculate that this is due to nonlinearity in the mapping from latent variable to spiking activity, as doubling the latent field ${\displaystyle h}$ does not double the population activity, but doubling the rate of a homogeneous Poisson process does double the expected spike count. As biological networks are likely to have such nonlinearities in their responses to common inputs, this scenario may be more applicable to certain kinds of recordings.

Latent variables – whether they are emergent from network dynamics (Clark et al., 2023; Sederberg and Nemenman, 2020) or derived from shared inputs – are ubiquitous in large-scale neural population recordings. This fact is reflected most directly in the relatively low-dimensional structure in large-scale population recordings (Stringer et al., 2019; Pandarinath et al., 2018; Nieh et al., 2021). We previously used a model based on this observation to examine signatures of neural criticality under a coarse-graining analysis and found that coarse-grained criticality is generated by systems driven by many latent variables (Morrell et al., 2021). Here, we showed that the same model also generates avalanche criticality, and that when information about the latent variables can be inferred from the network, avalanche criticality is also observed. Crucially, finding signatures of avalanche criticality required long observation times, such that the latent variable was well-sampled. Previous studies showed that Zipf’s law appears generically in systems coupled to a latent variable that changes slowly relative to the sampling time, and that the Zipf’s behavior is easier to observe in the higher information regime (Schwab et al., 2014; Aitchison et al., 2016). However, this also suggests that observation of either scaling at modest data set sizes indeed points to some fine-tuning — namely to the increase of the information in the individual neurons (and, since neurons in these models are conditionally independent, also in the entire network) about the value of the latent variables. In other words, one would expect a sensory part of the brain, if adapted to the statistics of the external stimuli, to exhibit all of these critical signatures at relatively modest data set sizes. In monocular deprivation experiments, when the activity in the visual cortex is transiently not adapted to its inputs, scaling disappears, at least for recordings of a typical duration, and is restored as the system adapts to the new stimulus (Ma et al., 2019). We speculate that the observed recovery of criticality by Ma et al., 2019 could be driven by neurons adapting to the reduced stimuli state, for instance, by adjusting ${\displaystyle \eta }$ (input scaling) and ${\displaystyle ϵ}$ (firing rate threshold).

Taken together, these results suggest that critical behavior in neural systems – whether based on the Zipf’s law, avalanches, or coarse-graining analysis – is expected whenever neural recordings exhibit some latent structure in population dynamics and this latent structure can be inferred from observations of the population activity.

We study a model from Morrell et al., 2021, originally constructed as a model of large populations of neurons in mouse hippocampus. In the original version of the model, neurons are non-interacting, receiving inputs reflective of place-field selectivity as well as input current arising from a random projection from a small number of latent dynamical variables, representing inputs shared across the population of neurons that are not directly measured or controlled. In the current paper, we incorporate only the latent variables (no place variables), and we assume that every cell is coupled to every latent variable with some randomly drawn coupling strength.

The probability of observing a certain population state ${\displaystyle \{s_i\}}$ given latent variables ${\displaystyle h_{\mu }(t)}$ at time ${\displaystyle t}$ is

where ${\displaystyle Z}$ is the normalization, and ${\displaystyle H}$ is the ‘energy’:

The latent variables ${\displaystyle h_{\mu }(t)}$ are Ornstein-Uhlenbeck processes with zero mean, unit variance, and time constant ${\displaystyle {\tau }_m}$. Couplings ${\displaystyle J_{i\mu }}$ are drawn from the standard normal distribution.

Parameters for each figure are laid out in Tables 1–3. For the infinite time constant simulation, we draw a value ${\displaystyle h\sim \mathcal{N}(0,1)}$ and simulate for 10000 time steps, then repeat for 1000 draws of ${\displaystyle h}$.

Most results were presented using arbitrary time units: all times (i.e. ${\displaystyle {\tau }_F}$ and avalanche duration ${\displaystyle D}$) are measured in units of an unspecified time step. Specifying a time bin converts the probability of firing into actual firing rates, in spikes per second, and this choice determines which part of the ${\displaystyle \eta }$-${\displaystyle ϵ}$ phase space is most relevant to a given experiment.

The time step is the temporal resolution at which activity is discretized, which varies from several to hundreds of milliseconds across different experimental studies (Beggs and Plenz, 2003; Fontenele et al., 2019; Ma et al., 2019). In physical units and assuming a bin size of 3 ms to 10 ms, our choice of ${\displaystyle \eta }$ and ${\displaystyle ϵ}$ in Figure 2 would yield physiologically realistic firing rate ranges (Hengen et al., 2016), with high-firing neurons reaching averages rates of 20-50 spikes/second and median firing-rate neurons around 1-2 spikes/second. The timescales of latent variables examined range from about 3 s to 3000 s, assuming 3-ms bins. Inputs with such timescales may arise from external sources, such as sensory stimuli, or from internal sources, such as changes in physiological state.

Simulations were carried out for the same number of time steps (2×10⁶), which would be approximately 1 to 2 ‘hours’, a reasonable duration for in vivo neural recordings. Note that at large values of ${\displaystyle {\tau }_F}$, the latent variable space is not well sampled during this time period.

Simulation code was adapted from our previous work (Morrell et al., 2021). Code to run simulations and perform analyses presented in this paper is uploaded as Source code 1 and also available from https://github.com/ajsederberg/avalanche (copy archived at Sederberg, 2024).

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

IN was supported in part by the Simons Foundation Investigator program, the Simons-Emory Consortium on Motor Control, NSF grant BCS/1822677 and NIH grant 2R01NS084844. AS was supported in part by NIH grant 1RF1MH130413-01 and by startup funds from the University of Minnesota Medical School. The authors acknowledge the Minnesota Supercomputing Institute (MSI) at the University of Minnesota for providing resources that contributed to the research results reported within this paper.

1. Sent for peer review: May 30, 2023
2. Preprint posted: June 8, 2023 (view preprint)
3. Preprint posted: September 12, 2023 (view preprint)
4. Preprint posted: February 2, 2024 (view preprint)
5. Version of Record published: March 12, 2024 (version 1)

You can cite all versions using the DOI https://doi.org/10.7554/eLife.89337. This DOI represents all versions, and will always resolve to the latest one.

© 2023, Morrell et al.

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