Mean-field theory of a plastic network of integrate-and-fire neurons

We consider a noise-driven network of integrate-and-fire neurons. The network evolves as result of the activities of the neurons following spike-timing-dependent plasticity rules. We apply a self-consistent mean-field theory to the system to obtain the mean activity level for the system as a function of the mean synaptic weight, which predicts a first-order transition and hysteresis between a noise-dominated regime and a regime of persistent neural activity. Assuming Poisson firing statistics for the neurons, the plasticity dynamics of a synapse under the influence of the mean-field environment can be mapped to the dynamics of an asymmetric random walk in synaptic-weight space. Using a master equation for small steps, we predict a narrow distribution of synaptic weights that scales with the square root of the plasticity rate for the stationary state of the system given plausible physiological parameter values describing neural transmission and plasticity. The dependence of the distribution on the synaptic weight of the mean-field environment allows us to determine the mean synaptic weight self-consistently. The effect of fluctuations in the total synaptic conductance and plasticity step sizes are also considered. Such fluctuations result in a smoothing of the first-order transition for low number of afferent synapses per neuron and a broadening of the synaptic-weight distribution, respectively.

Figure 1

(Color online) Firing frequency $\lambda \left(G\right)$ of an integrate-and-fire neuron under constant total synaptic conductance $G$ given by Eq. (16). Also shown are mean-field active synaptic transmitter fractions $Y\left(\lambda \right)$ (for $K=31$, $\overline{w}=0.01$) driven by a Poisson spike train of frequency $\lambda$ given by Eq. (17). The dashed line represents a numerically calculated correction to $Y\left(\lambda \right)$ due to the periodic firing of the neuron when driven by constant total synaptic conductance $G$. The parameters are listed in Table . The arrow indicates that the curve shifts to the right for increasing $\overline{w}$. The dot-dashed line is the neuron response function numerically calculated for the Morris-Lecar model used in [34] with a slightly higher background current.

Figure 2

(Color online) Mean-field firing frequency $\overline{\lambda }$ as a function of mean-field synaptic weight $\overline{w}$ determined by the intersections of neuron response function and synapse response function as plotted in Fig. 1 for $K=31$. The two arrows show the region of hysteresis where two stable fixed points, the noise-dominating lower fixed point and the persistently active upper fixed point, coexist. The symbols will be used in connection with Figs. 4, 7.

Figure 3

(Color online) We consider the dynamics of a single synapse of synaptic weight $w$ in a mean-field environment.

Figure 4

(Color online) Average synaptic weight from the simulation of a single synapse between two Poisson neurons as shown schematically in Fig. 3 versus the mean-field synaptic weight $\overline{w}$ for $w^{\star }=0.02$. The mean-field firing frequencies $\overline{\lambda }$ used in the single synapse simulations are given by Fig. 2, and the results for both the upper and lower fixed points (marked by corresponding symbols in Fig. 2) are plotted.

Figure 5

(Color online) Extended space of Fig. 1 when the fluctuation in synaptic conductance $G$ is considered (represented by the jump size $J$). The extended neuron response function (38), the synapse response function $Y\left(\overline{\lambda }\right)=\overline{G}/\left(K\overline{w}\right)$, and relation (39) determine the stationary states of a mean-field network (marked by a circle on the surfaces).

Figure 6

(Color online) Numerical mean-field phase diagram (firing frequency $\overline{\lambda }$ versus synaptic weight $\overline{w}$) for various numbers $K$ (as labeled in legend) of afferent synapses per neuron under fluctuating total synaptic conductance (drawn lines) compared with the analytical results when the total synaptic conductance $G=\overline{G}$ is assumed to be constant (dashed line).

Figure 7

(Color online) Numerical results (symbols) for the width of synaptic-weight distribution, $\Delta w$, as a function of plasticity rate, $r$, in a limited system of a single synapse between two neurons with Poisson spike trains following dynamics (11) and operating at three of the fixed points predicted by the mean-field theory (as marked by corresponding symbols in Fig. 2). The results are compared with the analytical predictions (lines of corresponding colors) from Eq. (42) (drawn lines) and Eq. (28) (dashed line).