Firing rate distributions in a feedforward network of neural oscillators with intrinsic and network heterogeneity (Q2091893)

In an earlier paper \textit{C. Ly} [J. Comput. Neurosci. 39, No. 3, 311--327 (2015; Zbl 1382.92067)], showed how the relationship between intrinsic and network heterogeneity alter firing rate distributions in a generic recurrent network model of leaky integrate-an-fire (LIF) neurons. In the present paper one considers feedforward networks of heterogeneous Morris-Lecar neurons and intends to obtain similar results. The second section starts with a description of the general feedforward oscillator model. The mathematical models are assumed to be Ito stochastic differential equations. One describes the phase reduction method to be applied to the model, one introduces the phase resetting curve or PCR of the neuron and one explains the meaning of two parameter: parameter $\Delta _ j$ that models the intrinsic heterogeneity and parameter $q_j$ , that models the network heterogeneity. The experiments are done on the multi-dimensional model of \textit{C. Morris} and \textit{H. Lecar} [``Voltage oscillations in the barnacle giant muscle fiber'', Biophys. J. 35, No. 1, 193--213 (1981; \url{doi:10.1016/S0006-3495(81)84782-0})]. The model is \[ C_m\frac{dV_j}{dt}=I_{pp} -g_L(V_j-E_L)-g_KW(t)(V_j-E_K)- g_{Ca}m_{\infty}(V_j)(V_j-E_{Ca})-q_j\sum_{k=1}^{M}w_{j,k}s_k(t)(V_j- E_{syn_k}) - \zeta\xi_j(t) \] \[ \frac{dW_j}{dt}=\phi\frac{W_{\infty}-W_j}{\tau_W} \] where \[m_{\infty}(V) = \frac{1}{2}[1+\tanh((V-V_1)/V_2)] \] \[ \tau_W(V) = 1/[\cosh((V-V_3)/(2V_4)), \] \[ W_{\infty}(V)=\frac{1}{2}[1+\tanh((V-V_3)/V_4)] \] All parameters are fixed except $V_3$, $V_4$, $\phi$, and the background current $I_{pp}$. By varying these four parameters one models the intrinsic heterogeneity resulting in different PRCs. The parameter $q_j$ from above, models the network heterogeneity. The synapse variable $s_k$(t) is modeled by a system of ordinary differential equations. To the feed-forward Morris-Lecar network the general phase reduction method is applied and one concentrates on the asymptotic approximation to the firing rate distribution. Experiments for different parameter sets are performed, results are plotted and largely discussed and one shows how relationship between the two forms of heterogeneity lead to significant changes in firing rate heterogeneity.