RETRACTED: Dynamics of Stochastic Integrate-and-Fire Networks

The neural dynamics generating sensory, motor, and cognitive functions are commonly understood through field theories for neural population activity. Classic neural field theories are derived from highly simplified models of individual neurons, while biological neurons are highly complex cells. Integrate-and-fire models retain a key nonlinear feature of neuronal activity: Action potentials return the membrane potential to a nearly fixed reset value. This nonlinear reset of the membrane voltage after a spike is absent from classic neural field theories. Here, we develop a statistical field theory for networks of integrate-and-fire neurons with stochastic spike emission. This reveals a new mean-field theory for the activity in these networks, fluctuation corrections to the mean-field dynamics, and a mapping to a self-consistent renewal process. We use these to study the impact of the spike-driven reset of the membrane voltage on population activity. The spike reset gives rise to a multiplicative, rate-dependent leak term in the mean-field membrane voltage dynamics. This leads to bistability between quiescent and active states in the mean-field theory of homogenous and excitatory-inhibitory pulse-coupled networks. We uncover two types of fluctuation corrections to the mean-field theory, due to the nonlinear mapping from membrane voltage to spike emission and the nonlinear reset. These two fluctuation corrections can have competing effects, promoting and suppressing activity, respectively. We then examine the roles of spike resets and recurrent inhibition in stabilizing network activity. We calculate the phase diagram for inhibitory stabilization and find that an inhibition-stabilized regime occurs in wide regions of parameter space, consistent with experimental reports of inhibitory stabilization in diverse brain regions. Fluctuations narrow the region of inhibitory stabilization, consistent with their role in suppressing activity through spike resets.

Popular Summary

Sensory, motor, and cognitive functions arise from the concerted activity of populations of neurons. Neuroscientists describe the dynamics of those populations using equations called neural field theories. The classic neural field theories were derived decades ago, using highly simplified model neurons. Neurons are, however, rich and complicated cells. The discrepancy between the complex biophysics of single neurons and the simplified assumptions underlying population-level models could be a major impediment to understanding neural function and dysfunction. In a step toward bridging this gap, we develop a framework for incorporating biophysical nonlinearities into neural field theories.

We study stochastic integrate-and-fire neurons, which model the nonlinear dynamics of an action potential by a stochastic spike-and-reset rule for the membrane voltage. Using tools from statistical field theory, we construct the joint probability of membrane voltages and spike trains in a network. This exposes a new, analytically tractable mean-field theory for the activity, as well as tools to study the impact of spiking fluctuations on average activity levels. We then use these tools to study the dynamics of simple model networks, deriving their phase diagrams and examining the roles of spiking fluctuations and recurrent synaptic feedback in promoting or suppressing activity.

This methodology applies to a broad class of models and may help construct a new bridge between the nonlinear biophysics of neurons and the emergent behavior of nervous systems.

Figure 1

Impact of fluctuations on firing rates through spike resets and nonlinear intensity functions. (a) Membrane voltage traces of the stochastic LIF neuron (top, black) and a neuron with linear resets (bottom, blue). For comparison, in this panel only the two neurons are forced to have the same spike times. (b) Firing rate vs resting voltage $E$ for three models. Black: the stochastic LIF neuron with a threshold-linear-intensity function $f(v)=\lfloor v-1{\rfloor }_{+}$. Blue: the linear-reset model with a matched intensity function $f_L(v)=f(v)$. Orange: the linear-reset model with matched mean-field membrane voltages $f_M(v)=vf(v)$. Dots: simulation. Solid curves: mean-field predictions [Eqs. (7), (10), (11)]. (c) Impact of fluctuations on the stochastic LIF neuron’s firing rate. Dotted: mean-field prediction. Dashed: self-consistent one-loop prediction accounting for Gaussian fluctuations around the expected voltage and rate [Eq. (12)]. Solid: exact renewal theory prediction from Eq. (17). (d) Difference between the one-loop and mean-field rates for the three models.

Figure 2

Bistable activity in homogenous networks. (a) Phase diagram of the mean-field theory Eq. (24) in the input ($E$) vs coupling ($J$) plane. There are three possible states: low activity ($L$), high activity ($H$), and bistability ($B$). (b)–(d) Raster plots of a homogenous network’s activity at the parameter locations marked in panel (a). At $t=5$ and $t=15$, perturbations of amplitude 2 and duration 2 are applied to the drive $E$ (top). (e) Bifurcation curve in $J$ with $E=1/2$. (f) Bifurcation plot in $E$ with $J=4$. Gray circles: simulation. Black dashed: the mean-field theory of Eq. (24). Black solid: the exact rate of the disorder-averaged system, using the numerical self-consistent solution of Eq. (23). The simulated network has Erdő-Rényi connectivity. Simulated network parameters: $N=100$, $p=0.5$. All nonzero connections have the same weight $J/(pN)$.

Figure 3

Bistable activity in excitatory-inhibitory networks. (a) Network diagram. (b) Phase diagram in the input ($E$) vs inhibitory strength ($g$) plane with $J=6$. There are three possible states: low activity ($L$), high activity ($H$), and bistability ($B$). (c) Phase diagram of the two-dimensional mean-field theory Eq. (32) in the input ($E$) vs coupling strength ($J$) plane with $g=1$. (d) Phase diagram in the coupling vs inhibitory strength plane with $E=0.5$. (e),(f) Example simulations with $(J,g)=(6,0.3)$, with $E=-0.5$ (e) or $E=0.5$ (f). The network has a block-Erős-Rényi structure. Simulated network parameters: population sizes $(N_e,N_i)=(200,50)$, excitatory output connection probabilities $p_{ee}=p_{ie}=0.2$, inhibitory output connection probabilities $p_{ei}=p_{ii}=0.8$. Within each block, all nonzero connections have the same weight, e.g., $J/(N_e{p}_e)$ for nonzero excitatory projections.

Figure 4

Fluctuations in an excitatory-inhibitory network with symmetric external inputs $E_e=E_i=E$. (a) Spike-train power spectrum with $(J,g,E)=(6,0.3,1.2)$. Dots: simulation of a network with 200 excitatory and 50 inhibitory neurons (population-averaged power spectrum). Dotted: the perturbative tree-level approximation (expanded around the deterministic mean-field theory). Dashed: the tree-level approximation around the one-loop rates. Solid: the renewal prediction of Eq. (37). (b) Interspike interval density with $(J,g,E)=(6,0.3,1.2)$. Dots: simulation. Solid: the renewal prediction of Eq. (36). (c),(d) Bifurcation diagrams for firing rate as a function. (e),(f) Bifurcation diagrams for spike-train variance. In (c) and (e), $(J,E)=(6,0.5)$. In (d) and (f), $(J,g)=(6,0.25)$. Simulated network parameters: population sizes $(N_e,N_i)=(200,50)$, excitatory output connection probabilities $p_{ee}=p_{ie}=0.5$, inhibitory output connection probabilities $p_{ei}=p_{ii}=0.8$.

Figure 5

Paradoxical responses to inhibitory stimulation. (a) Excitatory-inhibitory network with asymmetric drive. (b) Phase diagram and nullclines of the excitatory (blue) and inhibitory (orange) firing rates for the excitatory-inhibitory network with threshold-linear-rate functions. (c) Simulation of a block-Erdős-Rényi network with $p_{ee}=p_{ie}=0.5$, $p_{ei}=p_{ii}=0.8$. At time 0, $(E,h)=(2,1)$. At times 50 and 100, $h$ increases by $\frac{3}{4}$. Orange: inhibitory population-averaged spike train smoothed with a Gaussian kernel of width 2 for visualization. Parameters for (b) and (c): $(J,g,E)=(6,1/2,2)$. Simulated block Erdős-Rényi network parameters as in Fig. 3.

Figure 6

Phase diagrams for paradoxical responses to inhibitory stimulation. (a) Boundaries of the paradoxical response region with $J=4$. Dashed line: mean-field theory Eq. (43). Solid line: one-loop theory Eq. (45). Color: simulation. Each simulation lasts for 200 time units; at time 100, the inhibitory drive switches from $hE=E$ to $hE=E+0.1$. (b) As in (a), with $g=2$. Simulated block-Erdős-Rényi network parameters as in Fig. 3.