Impact of heavy-tailed synaptic strength distributions on self-sustained activity in networks of spiking neurons

Abstract

We analyze states of stationary activity in randomly coupled quadratic integrate-and-fire neurons using stochastic mean-field theory. Specifically, we consider the two cases of Gaussian random coupling and Cauchy random coupling, which are representative of systems with light- or with heavy-tailed synaptic strength distributions. For both, Gaussian and Cauchy coupling, bistability between a low activity and a high activity state of self-sustained firing is possible in excitable neurons. In the system with Cauchy coupling we find analytically a directed percolation threshold, i.e., above a critical value of the synaptic strength, activity percolates through the whole network starting from a few spiking units only. The existence of the directed percolation threshold is in agreement with previous numerical results in the literature for integrate-and-fire neurons with heavy-tailed synaptic strength distribution. However, we have found that the transition can be continuous or discontinuous, depending on the excitatory-inhibitory imbalance in the network. Networks with Gaussian coupling and networks with Cauchy coupling and additional additive noise lack the percolation transition in the thermodynamic limit.

Figures (7)

• Figure 1: Qualitative comparison between the network noise with Gaussian and with Cauchy random synaptic weights. Short-time, sliding window filtered inputs $\Delta_\tau[z^{NW}_\ell(t)]$ (Eqs. \ref{['Eq:GaussDeltaTau']} and \ref{['Eq:CauchyDeltaTau']}, window width $\tau=10\,\Delta t=0.01$), each for a single excitable neuron ($a_0=-0.5$). Taken from direct simulation of $N=1000$ QIF neurons (a) with Gaussian coupling disorder and (c) with Cauchy coupling disorder. The distribution of these values are shown as horizontal, normalized histograms to the right. Imbalance and heterogeneity of the synaptic weights in both examples are $\mu=4$ and $\sigma=4$ and we have not included additional noise to the neuron dynamics (see Figs.\ref{['Fig:GaussTheory_vs_Sims']}b and \ref{['Fig:CauchyTheory_vs_Sims']}b). The red curves superimposed on the histograms are the theoretical distributions $\mathcal{N}(\mu r \tau,\sigma^2 r \tau)$ and $\mathcal{C}(\mu r \tau, \sigma r \tau)$ in the Gaussian and in the Cauchy case, using the measured average firing rates in the networks $r=0.41$ and $r=0.36$, respectively. In (b) and (d), below the time series of the linearly filtered noise, we have aligned the corresponding time series of the single neuron membrane potentials $v_\ell(t)$ with jumps clearly visible for Cauchy distributed synaptic strengths. The dashed horizontal lines mark the resting potential and the saddle at $\pm\sqrt{-a_0}$, respectively. The working point of the neuron has shifted in both cases to positive values $a_0+\mu r>0$, i.e.,the self-sustained activity is mean-driven in this example.
• Figure 2: Input-output relation of pre- and postsynaptic firing rates. Input-output relations Eq.\ref{['Eq:PrePostMap']} for the rates of excitable QIF neurons with $\mu=0$, (a) heterogeneity $\sigma=3.5$ and different values of additive noise strength $D^{add}$ and (b) $D^{add}=0.08$ and different values of $\sigma$. Shown is the difference between the post- and the presynaptic firing rates, in the Gaussian white-noise approximation, when the postsynaptic neurons are driven by independent Poissonian presynaptic spike trains with firing rates ${r}^{pre}$. Stable self-consistent rates are located where the difference crosses zero with a negative slope smaller than two. There, a small deviation of ${r}^{pre}$ from that solution will result in a ${r}^{post}$ closer to the solution.
• Figure 3: Bistability region. We construct bifurcation diagrams by solving the self-consistency equation Eq. \ref{['Eq:RateConsistency']} parametrically (see Appendix \ref{['secA1']}). Shown in (a) is the self-consistent firing rate as a function of the additive noise strength for excitable neurons $a_0=-0.5$, coupling heterogeneity $\sigma=3.5$ and zero imbalance $\mu=0$. A stable state of high self-sustained activity co-exists with a stable state of low activity (solid lines), sustained by the additive noise over an interval of bistability. These two solutions are connected by an unstable solution branch (dashed line) and are created or annihilated in saddle-node bifurcations at the knees of the solution curve. With two control parameters, here shown in (b) for the heterogeneity $\sigma$ and the additive noise strength $D^{add}$, the surface of self-consistent firing rates has the form of a fold. Bistability exists between the saddle-node bifurcation lines (blue) which connect in a cusp point. The crosses correspond to points in the parameter space where we show the input-output relation for the pre- and postsynaptic firing rate in Fig.\ref{['Fig:PrePostMap']}. The dotted vertical line marks a cut at $\sigma=3.5$ across the bistability region which we have used in panel (a). The vertical lines in (a) correspond to the crosses in (b) at $\sigma=3.5$ and additive noise strengths $D^{add}\in\{0.04,0.08,0.1\}$, i.e., the input-output relations shown in Fig.\ref{['Fig:PrePostMap']}a.
• Figure 4: Theory and simulations for Gaussian random coupling. Shown are the self-consistent average firing rates, i.e., the neural activity, as functions of the heterogeneity $\sigma$ under the Gaussian white-noise approximation. We assume Gaussian random coupling Eq.\ref{['Eq:GaussianJ']} and different values of mean synaptic strength $\mu$. Dashed lines correspond to unstable, and solid lines to stable solutions. The QIF neurons are excitable with $a_0=-0.5$. In (a) and (b) the noise is purely self-generated, without additional noise, i.e., $D^{add}=0$, whereas in (c) and (d) we add additional independent Gaussian white noise of strength $D^{add}=0.1$ to each neuron. In (b) and (d) we compare the theoretical stationary activity, from the mean-field dynamics, with simulations of $N=1000$ neurons and excitatory imbalance $\mu=4$. Shown are averages and one standard deviation errorbars over ten forward (open circles) and backward scans (black circles) with different realizations of $J$ (fixed $\mu$ and adjusted $\sigma$) and over $T=100$ time units for each value of $\sigma$. Without noise, the inactive state in (b) is always stable in the limit $N\to\infty$ with very small synaptic strength.
• Figure 5: Bistability region. In case of Cauchy coupling, we obtain $\sigma=\sigma(r)$ in Eqs. \ref{['Eq:Cauchy_vbar_of_r']} and \ref{['Eq:Cauchy_sigma_of_r']} by solving the mean-field equations \ref{['Eq:QIFMF1']} and \ref{['Eq:QIFMF2']} for stationary solutions and plot the resulting bifurcation diagram in (a) for $a_0=-0.5$, $\mu=4.0$ and additive Cauchy noise of strength $\Gamma^{add}=0.21$. In (b) we plot $\sigma=\sigma(r)$ against $\Gamma^{add}(r)$ from Eq. \ref{['Eq:Cauchy_SGMadd_of_r']}. This parametric curve marks the saddle-node bifurcation of stable and unstable equilibria, i.e., the boundary of the bistability region. The additive noise strength in (a) was chosen at the height of the cusp point (dashed horizontal line in (b)), so that the low and the high activity states connect continuously with diverging slope (dashed vertical line in (a)) at a critical heterogeneity.