Effect of Inhibitory Spike-Timing-Dependent Plasticity on Fast Sparsely Synchronized Rhythms in A Small-World Neuronal Network

We consider the Watts-Strogatz small-world network (SWN) consisting of inhibitory fast spiking Izhikevich interneurons. This inhibitory neuronal population has adaptive dynamic synaptic strengths governed by the inhibitory spike-timing-dependent plasticity (iSTDP). In previous works without iSTDP, fast sparsely synchronized rhythms, associated with diverse cognitive functions, were found to appear in a range of large noise intensities for fixed strong synaptic inhibition strengths. Here, we investigate the effect of iSTDP on fast sparse synchronization (FSS) by varying the noise intensity $D$. We employ an asymmetric anti-Hebbian time window for the iSTDP update rule [which is in contrast to the Hebbian time window for the excitatory STDP (eSTDP)]. Depending on values of $D$, population-averaged values of saturated synaptic inhibition strengths are potentiated [long-term potentiation (LTP)] or depressed [long-term depression (LTD)] in comparison with the initial mean value, and dispersions from the mean values of LTP/LTD are much increased when compared with the initial dispersion, independently of $D$. In most cases of LTD where the effect of mean LTD is dominant in comparison with the effect of dispersion, good FSS (with higher spiking measure) is found to get better via LTD, while bad FSS (with lower spiking measure) is found to get worse via LTP. This kind of Matthew effect in inhibitory synaptic plasticity is in contrast to that in excitatory synaptic plasticity where good (bad) synchronization gets better (worse) via LTP (LTD). Emergences of LTD and LTP of synaptic inhibition strengths are intensively investigated via a microscopic method based on the distributions of time delays between the pre- and the post-synaptic spike times. Furthermore, we also investigate the effects of network architecture on FSS by changing the rewiring probability $p$ of the SWN in the presence of iSTDP.Comment: arXiv admin note: text overlap with arXiv:1704.0315

Burst Synchronization in A Scale-Free Neuronal Network with Inhibitory Spike-Timing-Dependent Plasticity

We are concerned about burst synchronization (BS), related to neural information processes in health and disease, in the Barab\'{a}si-Albert scale-free network (SFN) composed of inhibitory bursting Hindmarsh-Rose neurons. This inhibitory neuronal population has adaptive dynamic synaptic strengths governed by the inhibitory spike-timing-dependent plasticity (iSTDP). In previous works without considering iSTDP, BS was found to appear in a range of noise intensities for fixed synaptic inhibition strengths. In contrast, in our present work, we take into consideration iSTDP and investigate its effect on BS by varying the noise intensity. Our new main result is to find occurrence of a Matthew effect in inhibitory synaptic plasticity: good BS gets better via LTD, while bad BS get worse via LTP. This kind of Matthew effect in inhibitory synaptic plasticity is in contrast to that in excitatory synaptic plasticity where good (bad) synchronization gets better (worse) via LTP (LTD). We note that, due to inhibition, the roles of LTD and LTP in inhibitory synaptic plasticity are reversed in comparison with those in excitatory synaptic plasticity. Moreover, emergences of LTD and LTP of synaptic inhibition strengths are intensively investigated via a microscopic method based on the distributions of time delays between the pre- and the post-synaptic burst onset times. Finally, in the presence of iSTDP we investigate the effects of network architecture on BS by varying the symmetric attachment degree $l^*$ and the asymmetry parameter $\Delta l$ in the SFN.Comment: arXiv admin note: substantial text overlap with arXiv:1708.04543, arXiv:1801.0138

Effect of Network Architecture on Burst and Spike Synchronization in A Scale-Free Network of Bursting Neurons

We investigate the effect of network architecture on burst and spike synchronization in a directed scale-free network (SFN) of bursting neurons, evolved via two independent $\alpha-$ and $\beta-$processes. The $\alpha-$process corresponds to a directed version of the Barab\'{a}si-Albert SFN model with growth and preferential attachment, while for the $\beta-$process only preferential attachments between pre-existing nodes are made without addition of new nodes. We first consider the "pure" $\alpha-$process of symmetric preferential attachment (with the same in- and out-degrees), and study emergence of burst and spike synchronization by varying the coupling strength $J$ and the noise intensity $D$ for a fixed attachment degree. Characterizations of burst and spike synchronization are also made by employing realistic order parameters and statistical-mechanical measures. Next, we choose appropriate values of $J$ and $D$ where only the burst synchronization occurs, and investigate the effect of the scale-free connectivity on the burst synchronization by varying (1) the symmetric attachment degree and (2) the asymmetry parameter (representing deviation from the symmetric case) in the $\alpha-$process, and (3) the occurrence probability of the $\beta-$process. In all these three cases, changes in the type and the degree of population synchronization are studied in connection with the network topology such as the degree distribution, the average path length $L_p$, and the betweenness centralization $B_c$. It is thus found that not only $L_p$ and $B_c$ (affecting global communication between nodes) but also the in-degree distribution (affecting individual dynamics) are important network factors for effective population synchronization in SFNs.Comment: arXiv admin note: text overlap with arXiv:1504.0306

Effect of Spike-Timing-Dependent Plasticity on Stochastic Burst Synchronization in A Scale-Free Neuronal Network

We consider an excitatory population of subthreshold Izhikevich neurons which cannot fire spontaneously without noise. As the coupling strength passes a threshold, individual neurons exhibit noise-induced burstings. This neuronal population has adaptive dynamic synaptic strengths governed by the spike-timing-dependent plasticity (STDP). In the absence of STDP, stochastic burst synchronization (SBS) between noise-induced burstings of subthreshold neurons was previously found to occur over a large range of intermediate noise intensities. Here, we study the effect of additive STDP on the SBS by varying the noise intensity $D$ in the Barab\'asi-Albert scale-free network (SFN) for the case of symmetric preferential attachment. This type of SFN exhibits a power-law degree distribution, and hence it becomes an inhomogeneous one with a few "hubs" (i.e., super-connected nodes). Occurrence of a "Matthew" effect in synaptic plasticity is found to occur due to a positive feedback process. Good burst synchronization gets better via long-term potentiation (LTP) of synaptic strengths, while bad burst synchronization gets worse via long-term depression (LTD). Consequently, a step-like rapid transition to SBS occurs by changing $D$, in contrast to a relatively smooth transition in the absence of STDP. In the presence of additive STDP, we also investigate the effects of network architecture on the SBS for a fixed $D$. Emergences of LTP and LTD of synaptic strengths are investigated in details via microscopic studies based on both the distributions of time delays between the burst onset times of the pre- and the post-synaptic neurons and the pair-correlations between the pre- and the post-synaptic IIBRs (instantaneous individual burst rates). Finally, a multiplicative STDP case (depending on states) is also investigated in comparison with the additive STDP case (independent of states).Comment: arXiv admin note: substantial text overlap with arXiv:1704.0315

Emergence of Sparsely Synchronized Rhythms and Their Responses to External Stimuli in An Inhomogeneous Small-World Complex Neuronal Network

We consider an inhomogeneous small-world network (SWN) composed of inhibitory short-range (SR) and long-range (LR) interneurons. By varying the fraction of LR interneurons $p_{long}$, we investigate the effect of network architecture on emergence of sparsely synchronized rhythms, and make comparison with that in the Watts-Strogatz SWN. Although SR and LR interneurons have the same average in- and out-degrees, their betweenness centralities (characterizing the potentiality in controlling communication between other interneurons) are distinctly different. Hence, in view of the betweenness, SWNs we consider are inhomogeneous, unlike the "canonical" Watts-Strogatz SWN with nearly same betweenness centralities. As $p_{long}$ is increased, the average path length becomes shorter, and the load of communication traffic is less concentrated on LR interneurons, which leads to better efficiency of global communication between interneurons. Eventually, when passing a critical value $p_{long}^{(c)}$ $(\simeq 0.16)$, sparsely synchronized rhythms are found to emerge. We also consider two cases of external time-periodic stimuli applied to sub-groups of LR and SR interneurons, respectively. Dynamical responses (such as synchronization suppression and enhancement) to these two cases of stimuli are studied and discussed in relation to the betweenness centralities of stimulated interneurons, representing the effectiveness for transfer of stimulation effect in the whole network.Comment: arXiv admin note: text overlap with arXiv:1504.0306

Frequency-Domain Order Parameters for the Burst And Spike Synchronization Transitions of Bursting Neurons

We are interested in characterization of synchronization transitions of bursting neurons in the frequency domain. Instantaneous population firing rate (IPFR) $R(t)$, which is directly obtained from the raster plot of neural spikes, is often used as a realistic collective quantity describing population activities in both the computational and the experimental neuroscience. For the case of spiking neurons, a realistic time-domain order parameter, based on $R(t)$, was introduced in our recent work to characterize the spike synchronization transition. Unlike the case of spiking neurons, the IPFR $R(t)$ of bursting neurons exhibits population behaviors with both the slow bursting and the fast spiking timescales. For our aim, we decompose the IPFR $R(t)$ into the instantaneous population bursting rate $R_b(t)$ (describing the bursting behavior) and the instantaneous population spike rate $R_s(t)$ (describing the spiking behavior) via frequency filtering, and extend the realistic order parameter to the case of bursting neurons. Thus, we develop the frequency-domain bursting and spiking order parameters which are just the bursting and spiking "coherence factors" $\beta_b$ and $\beta_s$ of the bursting and spiking peaks in the power spectral densities of $R_b$ and $R_s$ (i.e., "signal to noise" ratio of the spectral peak height and its relative width). Through calculation of $\beta_b$ and $\beta_s$, we obtain the bursting and spiking thresholds beyond which the burst and spike synchronizations break up, respectively. Consequently, it is shown in explicit examples that the frequency-domain bursting and spiking order parameters may be usefully used for characterization of the bursting and the spiking transitions, respectively.Comment: arXiv admin note: substantial text overlap with arXiv:1403.399

Fast Sparsely Synchronized Brain Rhythms in A Scale-Free Neural Network

We consider a directed Barab\'{a}si-Albert scale-free network model with symmetric preferential attachment with the same in- and out-degrees, and study emergence of sparsely synchronized rhythms for a fixed attachment degree in an inhibitory population of fast spiking Izhikevich interneurons. For a study on the fast sparsely synchronized rhythms, we fix $J$ (synaptic inhibition strength) at a sufficiently large value, and investigate the population states by increasing $D$ (noise intensity). For small $D$, full synchronization with the same population-rhythm frequency $f_p$ and mean firing rate (MFR) $f_i$ of individual neurons occurs, while for sufficiently large $D$ partial synchronization with $f_p > {\langle f_i \rangle}$ ($\langle f_i \rangle$: ensemble-averaged MFR) appears due to intermittent discharge of individual neurons; particularly, the case of $f_p > 4 {\langle f_i \rangle}$ is referred to as sparse synchronization. Only for the partial and sparse synchronization, MFRs and contributions of individual neuronal dynamics to population synchronization change depending on their degrees, unlike the case of full synchronization. Consequently, dynamics of individual neurons reveal the inhomogeneous network structure for the case of partial and sparse synchronization, which is in contrast to the case of statistically homogeneous random graphs and small-world networks. Finally, we investigate the effect of network architecture on sparse synchronization in the following three cases: (1) variation in the degree of symmetric attachment (2) asymmetric preferential attachment of new nodes with different in- and out-degrees (3) preferential attachment between pre-existing nodes (without addition of new nodes). In these three cases, both relation between network topology and sparse synchronization and contributions of individual dynamics to the sparse synchronization are discussed.Comment: 54 pages, 13 figures. arXiv admin note: text overlap with arXiv:1403.103

Effect of Small-World Connectivity on Fast Sparsely Synchronized Cortical Rhythms

Fast cortical rhythms with stochastic and intermittent neural discharges have been observed in electric recordings of brain activity. Recently, Brunel et al. developed a framework to describe this kind of fast sparse synchronization in both random and globally-coupled networks of suprathreshold spiking neurons. However, in a real cortical circuit, synaptic connections are known to have complex topology which is neither regular nor random. Hence, in order to extend the works of Brunel et al. to realistic neural networks, we study the effect of network architecture on these fast sparsely synchronized rhythms in an inhibitory population of suprathreshold fast spiking (FS) Izhikevich interneurons. We first employ the conventional Erd\"{o}s-Renyi random graph of suprathreshold FS Izhikevich interneurons for modeling the complex connectivity in neural systems, and study emergence of the population synchronized states by varying both the synaptic inhibition strength $J$ and the noise intensity $D$. Thus, fast sparsely synchronized states of relatively high degree are found to appear for large values of $J$ and $D$. Second, for fixed values of $J$ and $D$ where fast sparse synchronization occurs in the random network, we consider the Watts-Strogatz small-world network of suprathreshold FS Izhikevich interneurons which interpolates between regular lattice and random graph via rewiring, and investigate the effect of small-world synaptic connectivity on emergence of fast sparsely synchronized rhythms by varying the rewiring probability $p$ from short-range to long-range connection. When passing a small critical value $p^*_c$ $(\simeq 0.12)$, fast sparsely synchronized population rhythms are found to emerge in small-world networks with predominantly local connections and rare long-range connections

Coupling-Induced Population Synchronization in An Excitatory Population of Subthreshold Izhikevich Neurons

We consider an excitatory population of subthreshold Izhikevich neurons which exhibit noise-induced firings. By varying the coupling strength $J$, we investigate population synchronization between the noise-induced firings which may be used for efficient cognitive processing such as sensory perception, multisensory binding, selective attention, and memory formation. As $J$ is increased, rich types of population synchronization (e.g., spike, burst, and fast spike synchronization) are found to occur. Transitions between population synchronization and incoherence are well described in terms of an order parameter $\cal{O}$. As a final step, the coupling induces oscillator death (quenching of noise-induced spikings) because each neuron is attracted to a noisy equilibrium state. The oscillator death leads to a transition from firing to non-firing states at the population level, which may be well described in terms of the time-averaged population spike rate $\overline{R}$. In addition to the statistical-mechanical analysis using $\cal{O}$ and $\overline{R}$, each population and individual state are also characterized by using the techniques of nonlinear dynamics such as the raster plot of neural spikes, the time series of the membrane potential, and the phase portrait. We note that population synchronization of noise-induced firings may lead to emergence of synchronous brain rhythms in a noisy environment, associated with diverse cognitive functions

Effect of Interpopulation Spike-Timing-Dependent Plasticity on Synchronized Rhythms in Neuronal Networks with Inhibitory and Excitatory Populations

We consider clustered small-world networks with both inhibitory (I) and excitatory (E) populations. This I-E neuronal network has adaptive dynamic I to E and E to I interpopulation synaptic strengths, governed by interpopulation spike-timing-dependent plasticity (STDP). In previous works without STDPs, fast sparsely synchronized rhythms, related to diverse cognitive functions, were found to appear in a range of noise intensity $D$ for static synaptic strengths. By varying $D$, we investigate the effect of interpopulation STDPs on diverse population and individual properties of fast sparsely synchronized rhythms that emerge in both the I- and the E-populations. Depending on values of $D$, long-term potentiation (LTP) and long-term depression (LTD) for population-averaged values of saturated interpopulation synaptic strengths are found to occur, and they make effects on the degree of fast sparse synchronization. In a broad region of intermediate $D$, the degree of good synchronization (with higher spiking measure) becomes decreased, while in a region of large $D$, the degree of bad synchronization (with lower spiking measure) gets increased. Consequently, in each I- or E-population, the synchronization degree becomes nearly the same in a wide range of $D$. This kind of "equalization effect" is found to occur via cooperative interplay between the average occupation and pacing degrees of fast sparsely synchronized rhythms. We note that the equalization effect in interpopulation synaptic plasticity is distinctly in contrast to the Matthew (bipolarization) effect in intrapopulation (I to I and E to E) synaptic plasticity where good (bad) synchronization gets better (worse). Finally, emergences of LTP and LTD of interpopulation synaptic strengths are intensively investigated via a microscopic method based on the distributions of time delays between the pre- and the post-synaptic spike times