Synchronization in Excitatory Networks with Medium Synaptic Delay

(1)

The first two terms in the right side of Eq. 1 are the ionic currents responsible for the generation of the action potential; the third term is the leakage current; the fourth term is the synaptic current; the fifth term stands for the external currents, which are different neuron by neuron. The detailed description of the first three terms can be found in [3]. The external drive for each neuron is drawn from a normal distribution with mean I_mean and standard deviation $\sigma$ . We assume that the synaptic current is the linear summation of each post-synaptic current component resulting from a single action potential. When neuron i receives a spike at time t_i, the post-synaptic current component is described by

${\mathrm{I}}_{\mathrm{i}}^{\mathrm{syn}}={\displaystyle {\sum}_{\mathrm{j}=1}^{\mathrm{N}}{\uprho}_{\mathrm{i},\mathrm{j}}\left(\mathrm{t}\right)\left[{\mathrm{v}}_{\mathrm{i}}\left(\mathrm{t}\right)-{\mathrm{E}}_{\mathrm{syn}}\right]}$

(2)

${\uprho}_{\mathrm{i},\mathrm{j}}\left(\mathrm{t}\right)=\left\{\begin{array}{l@{\quad}l}0 & {\mathrm{t}<\mathrm{tj}+\mathrm{d}}\\[8pt] {\mathrm{g}}_{\mathrm{syn}}\dfrac{1}{{}_{\tau }d{-}_{\tau }r}\left({\mathrm{e}}^{-\dfrac{\mathrm{t}-\mathrm{tj}-\mathrm{d}}{\tau^d}}\ -{\mathrm{e}}^{-\dfrac{\mathrm{t}-\mathrm{tj}-\mathrm{d}}{\tau^r}}\right) & {\mathrm{t}\ge \mathrm{tj}+{\mathrm{d}}^{\prime}}\end{array}\right.$

(3)

where E_syn is the reversal potential of the synapse; t_j is the recent firing time of neuron j; other four parameters in Eq. 3 are the synapse strength, rise time constant, decay time constant, and synaptic delay, respectively. If E_syn is high (for example 0 mV), the synapse is excitatory. The purpose of this paper is to search the parameter space to find the important parameters which support the robust synchronization. Synaptic delay is medium in this paper. If not stated otherwise, the parameter values in the network are set as follows:

${\mathrm{C}}_{\mathrm{m}}=1\;\upmu \mathrm{F}/\mathrm{c}{\mathrm{m}}^2,{\uptau}^{\mathrm{d}}=5\;\mathrm{ms},{\uptau}^{\mathrm{r}}=0.1\;\mathrm{ms},\mathrm{d}=5\;\mathrm{ms},\mathrm{N}=200.$

3 Robust Synchronization in Excitatory Networks

As the neurons connect with each other though synapses in networks, the robust network synchronization depends on the properties of the synaptic currents. We can see intuitively from Eq. 2 that the synaptic current difference between two neurons is sensitive to the difference of the membrane potential v_i(t) between two neurons, if reversal potential E_syn is low (Note that E_syn cannot be too low, as the synapse may be inhibitory if E_syn is too low) or synapse is strong. Consequently, the robust synchronization is sensitive to E_syn and synapse strength. Figure 1 shows that when reversal potential is low and synapse is strong, the neural firing in the network can be synchronized under a relatively high level of heterogeneity (the level of heterogeneity is characterized by the ratio of the standard deviation to the mean of the external drives). Figure 2 shows that the firing patterns in a network with medium reversal potential and medium synapse strength, where synchronization is less accurate than that in Fig. 1. When reversal potential is high and synapse strength is weak, the firing of neurons cannot be synchronized (see Fig. 3).