Responses to transient perturbation can distinguish intrinsic from latent criticality in spiking neural populations

Abstract

The critical brain hypothesis posits that neural circuitry operates near criticality to reap the computational benefits of accessing a wide range of timescales. The theory of critical phenomena generally predicts heavy-tailed (power-law) correlations in space and time near criticality, but it has been argued that in the brain such correlations could be inherited from ``latent variables,'' such as external sensory signals that are not directly observed when recording from neural circuitry. Distinguishing whether heavy-tailed correlations in neural activity are intrinsically generated within a neural circuit or are driven by unobserved latent variables is crucial for properly interpreting circuit functions. We argue that measuring neural responses to sudden perturbative inputs, rather than correlations in ongoing activity, can disambiguate these cases. We demonstrate this approach in a model of stochastic spiking neuron populations receiving external latent input that can be tuned to a critical state. We propose a scaling theory for the covariance and response functions of the spiking network, which we validate with simulations. We end by discussing how our approach might generalize to models of neural populations with more realistic biophysical details.

Figures (7)

• Figure 1: Spiking network model (A) A network of external inputs (blue) provides feedforward projections to the cortical network (green), which in turn generate discrete spikes (bottom). Inputs are correlated in space and time, representing spatiotemporal correlations in sensory input. The cortical neurons are recurrently connected, leading to spatiotemporal correlations due to both intrinsic circuit connections but also the correlations of the input. (B) The autocovariance of the spike train is given by the expectation of fluctuations around the mean at times $t,\,t'$: $C_{\dot{n}\dot{n}}(t) :=\mathbb{E}[\dot{n}(t+t_0)-\bar{n})(\dot{n}(t_0)-\bar{n})]$. The autocovariance $C_{\dot{n}\dot{n}}$ measures how similar a signal in time (dashed box) is to itself after after shifting the signal by $t$. The superimposed vertical lines show the shifting of the spike trains and the corresponding part of the covariance being calculated. (C) The response function $R_{\dot{n}V}(t)$ of the network is measured by perturbing the steady-state membrane potential by $\Delta V=5.0$ at a time $t_0$ (vertical cyan line) and measuring the average spike rate at a time $t+t_0$. When the network is an a steady state the covariances and a perturbation from the steady state are functions only of the time difference $t$.
• Figure 2: Spike covariance and response functions in a 25$^3$ lattice Each color bar denotes whether the given subsystem is close to criticality (red) or far from criticality (blue). (A) Spike-spike covariance functions corresponding to the combinations of the input and spiking network being near or far from their respective critical points. (B) The differential response of the average spike rate in response to a perturbation of size $\Delta V = \pm5$ to the membrane potential of each neuron. Dark gray boxes show the region over which the power law (green) was estimated; fits of the power-law exponential (gold) start from the left edge of the gray box and continue until the end of the trial. Spiking network: far-from-critical (blue) with $\xi_\text{spk}= 0.5$, near-critical (red) with $J_c - J=2\times10^{-4}$. Input network: far-from-critical (blue) with $r - r_c =1.0$, near-critical (red) with $r-r_c=10^{-3}$.
• Figure 3: Shape collapse of the covariance and response functions in a 25$^3$ lattice (A) Collapse of the spike-spike autocovariance in a network at criticality and without any input. (B) Shape collapse of the spike response functions in the same conditions as in (A). (C) Collapse of the spike response functions to a $\Delta V=5.0$ perturbation from the conditions in Fig. \ref{['fig:SpkGrid']}.
• Figure 4: Phases of neural activity in the spiking network model (A) Phase diagram of the spiking model outlined in Eqs. (\ref{['eqn:HawkesMembr']}-\ref{['eqn:HawkesSpk']}) on a 25$^3$ lattice. (B) Average firing rate of the spiking model along the bifurcation ridge in (A). (C)-(F) Example spike event rasters from various points along the bifurcation ridge.
• Figure 5: Quality of response collapse with number of trials The attempted collapse of response functions averaged over an increasing number of trials for the cases of: (Top) a near-critical spiking network ($J_c-J =0.0002$); (Bottom) a far-from-critical spiking ($J_c-J=0.5$).