Dynamical responses to external stimuli for both cases of excitatory and inhibitory synchronization in a complex neuronal network

Abstract

For studying how dynamical responses to external stimuli depend on the synaptic-coupling type, we consider two types of excitatory and inhibitory synchronization (i.e., synchronization via synaptic excitation and inhibition) in complex small-world networks of excitatory regular spiking (RS) pyramidal neurons and inhibitory fast spiking (FS) interneurons. For both cases of excitatory and inhibitory synchronization, effects of synaptic couplings on dynamical responses to external time-periodic stimuli S(t) (applied to a fraction of neurons) are investigated by varying the driving amplitude A of S(t). Stimulated neurons are phase-locked to external stimuli for both cases of excitatory and inhibitory couplings. On the other hand, the stimulation effect on non-stimulated neurons depends on the type of synaptic coupling. The external stimulus S(t) makes a constructive effect on excitatory non-stimulated RS neurons (i.e., it causes external phase lockings in the non-stimulated sub-population), while S(t) makes a destructive effect on inhibitory non-stimulated FS interneurons (i.e., it breaks up original inhibitory synchronization in the non-stimulated sub-population). As results of these different effects of S(t), the type and degree of dynamical response (e.g., synchronization enhancement or suppression), characterized by the dynamical response factor $D_f$ (given by the ratio of synchronization degree in the presence and absence of stimulus), are found to vary in a distinctly different way, depending on the synaptic-coupling type. Furthermore, we also measure the matching degree between the dynamics of the two sub-populations of stimulated and non-stimulated neurons in terms of a “cross-correlation” measure $M_c$. With increasing A, based on $M_c$, we discuss the cross-correlations between the two sub-populations, affecting the dynamical responses to S(t).

Appendix: Statistical-mechanical spiking measure in the stimulated and the non-stimulated sub-populations

We measure the degree of population synchronization in each of the stimulated and the non-stimulated sub-populations in terms of a realistic statistical-mechanical spiking measure, based on the ISPSR kernel estimate $R_s^{(l)}(t)$ ($l=1$ and 2 correspond to the stimulated and the non-stimulated cases, respectively) (Kim and Lim 2014). Population synchronization may be well visualized in the raster plot of spikes. For a synchronized case, spiking stripes or bursting bands (indicating population synchronization) appear successively in the raster plot, and the corresponding ISPSR kernel estimate $R_s^{(l)}(t)$ exhibits a regular oscillation. Each ith ($i=1,2,3,\ldots$) global cycle of $R_s^{(l)}(t)$ begins from its left minimum, passes the central maximum, and ends at the right minimum [also, corresponding to the beginning point of the next $(i+1)$th global cycle]; the 1st global cycle of $R_s^{(l)}(t)$ appears after transient times of $10^3\,\hbox {ms}$. Spikes which appear in the ith global cycle of $R_s^{(l)}(t)$ forms the ith stripe/band in the raster plot. To measure the degree of population synchronization in each of the stimulated ($l=1$) and the non-stimulated ($l=2$) sub-populations, a statistical-mechanical measure $M_s^{(l)}$, based on $R_s^{(l)}(t)$, was introduced by considering the occupation pattern and the pacing pattern of spikes in the stripes/bands (Kim and Lim 2014). The spiking measures $M_i^{(l)}$ of the ith stripe/band [appearing in the ith global cycle of $R_s^{(l)}(t)$] is defined by the product of the occupation degree $O_i^{(l)}$ of spikes (representing the density of the ith stripe/band) and the pacing degree $P_i^{(l)}$ of spikes (denoting the smearing of the ith stripe/band):

$$\begin{aligned} M_i^{(l)} = O_i^{(l)} \cdot P_i^{(l)}. \end{aligned}$$

(14)

The occupation degrees $O_i^{(l)}$ in the ith stripe/band is given by the fractions of spiking neurons in the ith stripe/band:

$$\begin{aligned} O_i^{(l)} = \frac{N_{i}^{(l)}}{N}, \end{aligned}$$

(15)

where $N_{i}^{(l)}$ is the number of spiking neurons in the ith stripe/band. For full synchronization with fully-occupied stripes/bands, $O_i^{(l)} = 1$, while for sparse synchronization with partially-occupied stripes/bands, $O_i^{(l)} < 1$. The pacing degree $P_i^{(l)}$ of spikes in the ith stripe/band can be determined in a statistical-mechanical way by taking into account their contributions to the macroscopic ISPSR kernel estimate $R_s^{(l)}(t)$. An instantaneous global phase $\varPhi ^{(l)}(t)$ of $R_s^{(l)}(t)$ was introduced via linear interpolation in the two successive subregions forming global cycles (Kim and Lim 2014). The global phase $\varPhi ^{(l)}(t)$ between the left minimum (corresponding to the beginning point of the ith global cycle) and the central maximum is given by

$$\begin{aligned} \varPhi ^{(l)}(t) = 2\pi (i-3/2) + \pi \left( \frac{t-t_{i}^{(l,min)}}{t_{i}^{(l,max)}-t_{i}^{(l,min)}} \right) \mathrm{\,\, for\,} \,t_{i}^{(l,min)} \le t < t_{i}^{(l,max)}, \end{aligned}$$

(16)

and $\varPhi ^{(l)}(t)$ between the central maximum and the right minimum [corresponding to the beginning point of the $(i+1)$th global cycle] is given by

$$\begin{aligned} \varPhi ^{(l)}(t) = 2\pi (i-1) + \pi \left( \frac{t-t_{i}^{(l,max)}}{t_{i+1}^{(l,min)}-t_{i}^{(l,max)}} \right) \mathrm{\,\, for\,} \,t_{i}^{(l,max)} \le t < t_{i+1}^{(l,min)}, \end{aligned}$$

(17)

where $t_{i}^{(l,min)}$ is the beginning time of the ith ($i=1, 2, 3, \ldots$) global cycle of $R_s^{(l)}(t)$ [i.e., the time at which the left minimum of $R_s^{(l)}(t)$ appears in the ith global cycle], and $t_{i}^{(l,max)}$ is the time at which the maximum of $R_s^{(l)}(t)$ appears in the ith global cycle. Then, the contributions of the kth microscopic spikes in the ith stripe/band occurring at the times $t_{k}^{(l)}$ to $R_s^{(l)}(t)$ is given by $\cos \varPhi _k^{(l)}$, where $\varPhi _k^{(l)}$ are the global phases at the kth spiking time [i.e., $\varPhi _k^{(l)} \equiv \varPhi ^{(l)}(t_{k}^{(l)})$]. Microscopic spikes make the most constructive (in-phase) contributions to $R_s^{(l)}(t)$ when the corresponding global phases $\varPhi _k^{(l)}$ is $2 \pi n$ ($n=0,1,2, \ldots$), while they make the most destructive (anti-phase) contribution to $R_s^{(l)}(t)$ when $\varPhi _k^{(l)}$ is $2 \pi (n-1/2)$. By averaging the contributions of all microscopic spikes in the ith stripe/band to $R_s^{(l)}(t)$, we obtain the pacing degrees $P_i^{(l)}$ of spikes in the ith stripe/band:

$$\begin{aligned} P_i^{(l)} = { \frac{1}{S_i^{(l)}}} \sum _{k=1}^{S_i^{(l)}} \cos \varPhi _k^{(l)} \end{aligned}$$

(18)

where $S_i^{(l)}$ is the total number of microscopic spikes in the ith stripe/band. By averaging $M_i^{(l)}$ of Eq. (14) over a sufficiently large number $N_s^{(l)}$ of stripes/bands, we obtain the statistical-mechanical spiking measure $M_s^{(l)}$:

$$\begin{aligned} M_s^{(l)} = {\frac{1}{N_s^{(l)}}} \sum _{i=1}^{N_s^{(l)}} M_i^{(l)}. \end{aligned}$$

(19)

Here, we follow $3 \times 10^3$ global cycles in each realization, and obtain the average occupation degree, the average pacing degree, and the average statistical-mechanical spiking measure via average over 30 (60) realizations for the case of the non-stimulated (stimulated) sub-population. Here, more realizations are necessary for the stimulated case because the number $N_s ({=}50)$ of stimulated neurons is much less than that (${=}950$) of non-stimulated neurons.