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Effect of Synaptic Plasticity on Brain Rhythms, Associated with Diverse Brain Functions and Diseases, in Complex Biological Neural Networks

(1) Effect of Intrapopulation Spike-Timing-Dependent Plasticity on Synchronized Rhythms in Complex Neural Networks

  We consider a complex neural network consisting of excitatory (E) or inhibitory (I) spiking or bursting neurons. This neuronal network has adaptive dynamic E to E or I to I intrapopulation synaptic strengths governed by the spike-timing-dependent plasticity (STDP). In previous works without STDP, synchronized population rhythm was found to occur in a range of intermediate noise intensities. We investigate the effect of additive STDP on the synchronized rhythm by varying the noise intensity. Occurrence of a "Matthew" effect in synaptic plasticity is found due to a positive feedback process. As a result, good synchronization gets better via long-term potentiation (LTP) of synaptic strengths, while bad synchronization gets worse via long-term depression (LTD). Furthermore, emergences of long-term potentiation and long-term depression of synaptic strengths are intensively investigated via microscopic studies based on the distributions of time delays between the pre- and the post-synaptic spike times.

[1] S.-Y. Kim and W. Lim, "Stochastic spike synchronization in a small-world neural network with spike-timing-dependent plasticity," Neural Networks 97, 92-106 (2018).
[2] S.-Y. Kim and W. Lim, "Effect of spike-timing-dependent plasticity on stochastic burst synchronization in a scale-free neuronal network," Cognitive Neurodynamics 12, 315-342 (2018).

[3] S.-Y. Kim and W. Lim, "Effect of inhibitory spike-timing-dependent plasticity on fast sparsely synchronized rhythms in a small-world neuronal network," Neural Networks 106, 50-66 (2018).

[4] S.-Y. Kim and W. Lim, "Burst synchronization in a scale-free neuronal network with inhibitory spike-timing-dependent plasticity," Cognitive Neurodynamics 13, 53-73 (2019).

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

 We consider a two-population network consisting of both inhibitory (I) interneurons and excitatory (E) pyramidal cells. 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. Here, by varying D, we investigate the effect of interpopulation STDPs on 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. Then, the degree of fast sparse synchronization varies due to effects of LTP and LTD. In a broad region of intermediate D, the degree of good synchronization (with higher synchronization degree) becomes decreased, while in a region of large D, the degree of bad synchronization (with lower synchronization degree) gets increased. Consequently, in each I- or E-population, the synchronization degree becomes nearly the same in a wide range of D (including both the intermediate and the large D regions). This kind of "equalization effect" is found to occur via cooperative interplay between the average occupation and pacing degrees of spikes (i.e., the average fraction of firing neurons and the average degree of phase coherence between spikes in each synchronized stripe of spikes in the raster plot of spikes) in fast sparsely synchronized rhythms. Finally, emergences of LTP and LTD of interpopulation synaptic strengths (leading to occurrence of equalization effect) are intensively investigated via a microscopic method based on the distributions of time delays between the pre- and the post-synaptic spike times.

[1] S.-Y. Kim and W. Lim, "Effect of interpopulation spike-timing-dependent plasticity on synchronized rhythms in neuronal networks with inhibitory and excitatory populations," Cognitive Neurodynamics 14, 535-567 (2020).

(3) Effect of Diverse Recoding of Granule Cells on Optokinetic Response and Delay Eyeblink Conditioning in A Cerebellar Ring Network with Synaptic Plasticity

 We develop a cerebellar ring network, and investigate the effect of diverse recoding of granule (GR) cells on optokinetic response (OKR associated with spatial motor control) and delay Pavlovian eyeblink conditioning (EBC related to temporal motor control) by varying the connection probability pc from Golgi to GR cells. For an optimal value of pc, individual GR cells exhibit diverse spiking patterns. In the case of OKR, they are in-phase, anti-phase, or complex out-of-phase with respect to their population averaged firing activity. In the case of EBC, individual granule cells show various well- and ill-matched firing patterns relative to the unconditioned stimulus. Then, these diversely-recoded signals are fed into the Purkinje cells (PCs) through parallel-fibers (PFs), and the instructor climbing-fiber (CF) signals from the inferior olive depress them effectively. For the OKR, synaptic weights at in-phase PF-Purkinje cell (PC) synapses of active GR cells are strongly depressed via strong long-term depression (LTD), while those at anti-phase and complex out-of-phase PF-PC synapses are weakly depressed through weak LTD. Similar synaptic plasticity also occurs for the EBC. In the case of well-matched PF-PC synapses, their synaptic weights are strongly depressed through strong long-term depression (LTD). On the other hand, practically no LTD occurs for the ill-matched PF-PC synapses. This kind of effective depression (i.e., strong/weak LTD) at the PF-PC synapses causes a big modulation in firings of PCs. Then, PCs exert effective inhibitory coordination on the vestibular nucleus (VN) neuron (which evokes OKR) doe the OKR and cerebellar nucleus neuron [which elicits conditioned response (CR)] for the eyeblink conditioning. In the case of OKR, for the firing of the VN neuron, the learning gain degree Lg, corresponding to the modulation gain ratio, increases with increasing the learning cycle, and it saturates. In the case of EBC, when the learning trial passes a threshold, acquisition of CR begins. In this case, the timing degree Td of CR becomes good due to presence of the ill-matched firing group which plays a role of protection barrier for the timing. With further increase in the trial, strength S of CR (corresponding to the amplitude of eyelid closure) increases due to strong LTD in the well-matched firing group, while its timing degree Td decreases. In this way, the well- and the ill-matched firing groups play their own roles for the strength and the timing of CR, respectively. Thus, with increasing the learning trial, the (overall) learning efficiency Le (taking into consideration both timing and strength of CR) for the CR is increased, and eventually it becomes saturated. By changing pc from its optimal value, we also investigate the effect of diverse recoding on the OKR and the EBC. It is thus found that the more diverse in recoding of GR cells, the more effective in motor learning for the OKR and the Pavlovian EBC.

[1] S.-Y. Kim and W. Lim, "Effect of diverse recoding of granule cells on optokinetic response in a cerebellar ring network with synaptic plasticity," e-print: arXiv:2003.11325[q-bio.NC]; bioRxiv: DOI:10.1101/2020.03.25.007245.
[2]
S.-Y. Kim and W. Lim, "Influence of various temporal recoding on Pavlovian eyeblink conditioning in the cerebellum," e-print: arXiv:2006.14933[q-bio.NC]; bioRxiv: DOI:10.1101/2020.06.23.168294.


Sparsely Synchronized Brain Rhythms in Complex Neural Networks

 Sparsely-synchronized brain rhythms, associated with diverse cognitive functions such as sensory perception, feature integration, selective attention, and memory formation, have been observed in electric recordings (e.g., EEG and local field potentials) of brain activity. At the population level, cortical rhythms exhibit small-amplitude fast oscillations, while at the cellular level individual neurons show stochastic firings sparsely at a much lower rate than the population rate. To resolve the apparent dichotomy between synchrony at the population level and stochasticity at the cellular level, we study emergence of sparsely synchronized brain rhythms in networks of Geiger-counter-like neurons exhibiting irregular discharges. Previously, sparse synchronization was found to occur for cases of both global coupling (i.e., regular all-to-all coupling) and random coupling. However, a real neural network is known to be non-regular and non-random because it has complex topology (e.g., small-worldness, scale-freeness, and modularity). Hence, we investigate the effect of the network architecture on emergence of sparsely synchronized rhythms in real complex networks such as the small-world, the scale-free, the star-like, and the clustered networks. 

(1) Sparsely Synchronized Brain Rhythm in A Small-World Neural Network

 Sparsely synchronized cortical rhythms, associated with diverse cognitive functions, have been observed in electric recordings of brain activity. At the population level, cortical rhythms exhibit small-amplitude fast oscillations, while at the cellular level, individual neurons show stochastic firings sparsely at a much lower rate than the population rate. We study the effect of network architecture on sparse synchronization in an inhibitory population of Geiger-counter-like neurons showing stochastic discharges. As an element of the network, we consider the subthreshold Morris-Lecar neuron and the fast spiking Izhikevich interneuron which fires as Geiger counters. When strong external noise balances with strong synaptic inhibition, sparsely synchronized rhythms are found to emerge. Previously, sparse synchronization was found to occur for both cases of global coupling (i.e., regular all-to-all coupling) and random coupling. However, a real neural network is known to be neither regular, nor random. Here, we consider sparse Watts-Strogatz small-world networks which interpolates between the regular lattice and the random graph via rewiring. We start from the regular lattice with only short-range connections, and then investigate emergence of sparse synchronization by increasing the rewiring probability p from short-range to long-range connections. For p=0, the average synaptic path length between pairs of neurons becomes long, and hence there exists only an unsynchronized population state because the global efficiency of information transfer is low. However, as p is increased, long-range connections begin to appear, and global effective communication between distant neurons may be available via shorter synaptic paths. Consequently, as p passes a threshold pth, sparsely synchronized population rhythms emerge. However, with increasing p longer axon wirings become expensive because of their material and energy costs. At an optimal value p*DE of the rewiring probability, the ratio of the synchrony degree to the wiring cost is found to become maximal. In this way, an optimal sparse synchronization is found to occur at a minimal wiring cost in an economic small-world network through trade-off between synchrony and wiring cost.

[1] S.-Y. Kim and W. Lim, "Sparsely-synchronized brain rhythm in a small-world neural network" J. Korean Phys. Soc. 63, 104-113 (2013).
[2]
S.-Y. Kim and W. Lim, "Effect of small-world connectivity on fast sparsely synchronized cortical rhythms," Physica A 421, 109-123 (2015).

(2) Sparsely Synchronized Brain Rhythm in A Scale-Free Neural Network

 We consider a directed version of the Barabasi-Albert scale-free network model with symmetric preferential attachment with the same in- and out-degrees and study the emergence of sparsely synchronized rhythms for a fixed attachment degree in an inhibitory population of fast-spiking Izhikevich interneurons. Fast sparsely synchronized rhythms with stochastic and intermittent neuronal discharges are found to appear for large values of J (synaptic inhibition strength) and D (noise intensity). For an intensive study we fix J at a sufficiently large value and investigate the population states by increasing D. For small D, full synchronization with the same population-rhythm frequency fP and mean firing rate (MFR) fi of individual neurons occurs, while for large D partial synchronization with fP > <fi> (<fi>: ensemble-averaged MFR) appears due to intermittent discharge of individual neurons; in particular, the case of fP > 4<fi> is referred to as sparse synchronization. For the case of partial and sparse synchronization, MFRs of individual neurons vary depending on their degrees. As D passes a critical value D* (which is determined by employing an order parameter), a transition to unsynchronization occurs due to the destructive role of noise to spoil the pacing between sparse spikes. For D<D*, population synchronization emerges in the whole population because the spatial correlation length between the neuronal pairs covers the whole system. Furthermore, the degree of population synchronization is also measured in terms of two types of realistic statistical-mechanical measures. Only for the partial and sparse synchronization do contributions of individual neuronal dynamics to population synchronization change depending on their degrees, unlike in 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 for fixed values of J and D 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, and (3) preferential attachment between pre-existing nodes (without addition of new nodes). In these three cases, both relation between network topology (e.g., average path length and betweenness centralization) and sparse synchronization and contributions of individual dynamics to the sparse synchronization are discussed.

[1] S.-Y. Kim and W. Lim, "Fast sparsely synchronized brain rhythms in a scale-free neural network," Phys. Rev. E 92, 022717 (2015).

(3) Sparsely Synchronized Brain Rhythm in Clustered Small-World Neural Networks

 We consider a clustered network with small-world sub-networks of inhibitory fast spiking interneurons, and investigate the effect of inter-modular connection on emergence of fast sparsely synchronized rhythms by varying both the inter-modular coupling strength Jinter and the average number of inter-modular links per interneuron M(inter)syn. In contrast to the case of non-clustered networks, two kinds of sparsely synchronized states such as modular and global synchronization are found. For the case of modular sparse synchronization, the population behavior reveals the modular structure, because the intra-modular dynamics of sub-networks make some mismatching. On the other hand, in the case of global sparse synchronization, the population behavior is globally identical, independently of the cluster structure, because the intra-modular dynamics of sub-networks make perfect matching. We introduce a realistic cross-correlation modularity measure, representing the matching-degree between the instantaneous sub-population spike rates of the sub-networks, and examine whether the sparse synchronization is global or modular. Depending on its magnitude, the inter-modular coupling strength Jinter seems to play dual roles for the pacing between spikes in each sub-network. For large Jinter, due to strong inhibition it plays a destructive role to spoil the pacing between spikes, while for small Jinter it plays a constructive role to favor the pacing between spikes. Through competition between the constructive and the destructive roles of Jinter, there exists an intermediate optimal Jinter at which the pacing degree between spikes becomes maximal. In contrast, the average number of inter-modular links per interneuron M(inter)syn seems to play a role just to favor the pacing between spikes. With increasing M(inter)syn, the pacing degree between spikes increases monotonically thanks to the increase in the degree of effectiveness of global communication between spikes. Furthermore, we employ the realistic sub- and whole-population order parameters, based on the instantaneous sub- and whole-population spike rates, to determine the threshold values for the synchronization-unsynchronization transition in the sub- and whole-populations, and the degrees of global and modular sparse synchronization are also measured in terms of the realistic sub- and whole-population statistical-mechanical spiking measures defined by considering both the occupation and the pacing degrees of spikes. It is expected that our results could have implications for the role of the brain plasticity in some functional behaviors associated with population synchronization.

[1] S.-Y. Kim and W. Lim, "Effect of inter-modular connection on fast sparse synchronization in clustered small-world neural networks," Phys. Rev. E 92, 052716 (2015).


Burst Synchronization in Complex Neural Systems

We are concerned about population synchronization of bursting neurons. Bursting occurs when neuronal activity alternates, on a slow timescale, between a silent phase and an active (bursting) phase of fast repetitive spikings. This type of bursting activity occurs due to the interplay of the fast ionic currents leading to spiking activity and the slower currents modulating the spiking activity. Hence, the dynamics of bursting neurons have two timescales: slow bursting timescale and fast spiking timescale. These bursting neurons exhibit two different patterns of synchronization due to the slow and the fast timescales of bursting activity. Burst synchronization (synchrony on the slow bursting timescale) refers to a temporal coherence between the active phase (bursting) onset or offset times of bursting neurons, while spike synchronization (synchrony on the fast spike timescale) characterizes a temporal coherence between intraburst spikes fired by bursting neurons in their respective active phases. We study burst and spike synchronization of bursting neurons, associated with neural information processes in health and disease, in complex networks such as small-world, scale-free, and clustered networks.

(1) Stochastic Bursting Synchronization in A Population of Subthreshold Izhikevich Neurons

 We are interested in neural bursting activity (alternating between a silent phase and an active phase of repetitive spiking). Cortical intrinsically bursting neurons, thalamocortical relay neurons, thalamic reticular neurons, and hippocampal pyramidal neurons are representative examples of bursting neurons. We investigate coherent population dynamics in these bursting neurons by varying the noise intensity D. Such coherence is well visualized in the raster plot of neural spikings. For a coherent state, burst bands, composed of stripes of spikes, constitute the raster plot. For this case, burst synchronization refers to a temporal relationship between active phase onset or offset times of bursting neurons, while spike synchronization characterizes a temporal relationship between spikes fired by different bursting neurons in their respective active phases. For the coherent case, in addition to burst synchronization, spike synchronization also occurs in each burst band. As a result of this complete synchronization, the (ensemble-averaged) global membrane potential VG exhibits a bursting activity like the local membrane potential (i.e., fast spikes appear on a slow wave in VG). However, as D is increased, loss of spike coherence first occurs in each burst band due to smearing of stripes of spikes. With further increase in D, overlapping between bands begins to occur, which eventually leads to complete loss of burst synchronization. To characterize these burst and spike coherence, we introduce a new type of coherence measure quantifying the degree of coherence seen in the raster plot of neural spiking, where the global potential is used to give a reference phase for the burst onset times and the spiking times in active phases.

[1] S.-Y. Kim, Y. Kim, D.-G. Hong, J. Kim, and W. Lim, "Stochastic bursting synchronization in a population of subthreshold Izhikevich neurons", J. Korean Phys. Soc. 60, 1441-1447 (2012).

(2) 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 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 R. In addition to the statistical- mechanical analysis using O and 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.

[1] S.-Y. Kim and W. Lim, "Coupling-induced population synchronization in an excitatory population of subthreshold Izhikevich neurons," Cognitive Neurodynamics 7, 495-503 (2013).

(3) Noise-Induced Burst And Spike Synchronizations in An Inhibitory Small-World Network of Subthreshold Bursting Neurons

 We are interested in noise-induced firings of subthreshold neurons which may be used for encoding environmental stimuli. Noise-induced population synchronization was previously studied only for the case of global coupling, unlike the case of subthreshold spiking neurons. Hence, we investigate the effect of complex network architecture on noise-induced synchronization in an inhibitory population of subthreshold bursting Hindmarsh–Rose neurons. For modeling complex synaptic connectivity, we consider the Watts–Strogatz small-world network which interpolates between regular lattice and random network via rewiring, and investigate the effect of small-world connectivity on emergence of noise-induced population synchronization. Thus, noise-induced burst synchronization (synchrony on the slow bursting time scale) and spike synchronization (synchrony on the fast spike time scale) are found to appear in a synchronized region of the J–D plane (J: synaptic inhibition strength and D: noise intensity). As the rewiring probability p is decreased from 1 (random network) to 0 (regular lattice), the region of spike synchronization shrinks rapidly in the J–D plane, while the region of the burst synchronization decreases slowly. We separate the slow bursting and the fast spiking time scales via frequency filtering, and characterize the noise-induced burst and spike synchronizations by employing realistic order parameters and statistical-mechanical measures introduced in our recent work. Thus, the bursting and spiking thresholds for the burst and spike synchronization transitions are determined in terms of the bursting and spiking order parameters, respectively. Furthermore, we also measure the degrees of burst and spike synchronizations in terms of the statistical-mechanical bursting and spiking measures, respectively.

[1] S.-Y. Kim and W. Lim, "Noise-induced burst and spike synchronizations in an inhibitory small-world network of subthreshold bursting neurons," Cognitive Neurodynamics 9, 179-200 (2015).

(4) 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 − and −processes. The −process corresponds to a directed version of the Barabasi-Albert SFN model with growth and preferential attachment, while for the −process only preferential attachments between preexisting nodes are made without addition of new nodes. We first consider the pure −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 −process, and (3) the occurrence probability of the −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 Lp, and the betweenness centralization Bc. It is thus found that not only Lp and Bc (affecting global communication between nodes) but also the in-degree distribution (affecting individual dynamics) are important network factors for effective population synchronization in SFNs.

[1] S.-Y. Kim and W. Lim, "Effect of network architecture on burst and spike synchronization in a scale-free network of bursting neurons," Neural Networks 79, 53-77 (2016).


Thermodynamic and Statistical-Mechanical Measures for Characterization of Neural Synchronization

 Synchronized brain rhythms in sleep and awake states (e.g., alpha, sleep spindle, gamma, ultrafast, beta rhythms), associated with diverse sensory and cognitive functions, have been observed in electrical recordings (EEG and local field potentials) of brain activity. Neural synchronization may be well described by using the population-averaged global potential VG in computational neuroscience, However, to obtain VG in real experiments is practically difficult. Instead of VG, the instantaneous population spike rate (IPSR), which may be obtained experimentally, is used for description of population behaviors in both computational and experimental neuroscience. The time-averaged fluctuation of IPSR plays the role of an order parameter O used for describing the synchrony-asynchrony transition in neural systems. This order parameter can be regarded as a thermodynamic measure because it concerns only the macroscopic IPSR without considering any relation between IPSR and microscopic individual potentials (spikes). Population spike synchronization may be well seen in the raster plot of neural spikes. The degree of collective spike synchronization seen in the raster plot may be well measured in terms of a new ``statistical-mechanical'' spike-based measure Ms introduced by considering the occupation and the pacing patterns of spikes in the raster plot. In particular, the pacing degree between spikes is determined in a statistical-mechanical way by quantifying the average contribution of (microscopic) individual spikes to the macroscopic IPSR. This ``statistical-mechanical'' measure Ms is in contrast to the conventional measures such as the ``thermodynamic'' order parameter (which concerns the time-averaged fluctuations of the macroscopic global potential), the ``microscopic'' correlation-based measure (based on the cross-correlation between the microscopic individual potentials), and the measures of precise spike timing [based on the peri-stimulus time histogram (PSTH)]. For the conventional PSTH-based measure ``events,'' corresponding to peaks of the IPSR, are selected through setting a threshold. Then, the measures for the reliability and the precision of spike timing concern only the spikes within the events, in contrast to the case of the ``statistical-mechanical'' measure where all spikes are considered (without selecting events). A main difference between the conventional and the new spike-based measures lies in determining the pacing degree of spikes. The precision of spike timing for the conventional case is given by just the standard deviation of (microscopic) individual spike times within an event without considering the quantitative contribution of (microscopic) individual spikes to the (macroscopic) global activity. Hence, the PSTH-based measure is not a statistical-mechanical measure. Finally, we emphasize that both the realistic thermodynamic and statistical-mechanical measures, based on the IPSR, may be practically applicable for characterization of neural synchronization in both the computational and the experimental neuroscience such as  spike-timing reliability, stimulus discrimination, and event related sync./desync.

(1) Statistical-Mechanical Measure of Stochastic Spiking Coherence in A Population of Inhibitory Subthreshold Neurons

 By varying the noise intensity, we study stochastic spiking coherence (i.e., collective coherence between noise-induced neural spikings) in an inhibitory population of subthreshold neurons (which cannot fire spontaneously without noise). This stochastic spiking coherence may be well visualized in the raster plot of neural spikes. For a coherent case, partially-occupied "stripes" (composed of spikes and indicating collective coherence) are formed in the raster plot. This partial occupation occurs due to "stochastic spike skipping" which is well shown in the multi-peaked interspike interval histogram. The main purpose of our work is to quantitatively measure the degree of stochastic spiking coherence seen in the raster plot. We introduce a new spike-based coherence measure Ms by considering the occupation pattern and the pacing pattern of spikes in the stripes. In particular, the pacing degree between spikes is determined in a statistical-mechanical way by quantifying the average contribution of (microscopic) individual spikes to the (macroscopic) ensemble-averaged global potential. This "statistical-mechanical" measure Ms is in contrast to the conventional measures such as the "thermodynamic" order parameter (which concerns the time-averaged fluctuations of the macroscopic global potential), the "microscopic" correlation-based measure (based on the cross-correlation between the microscopic individual potentials), and the measures of precise spike timing (based on the peri-stimulus time histogram). In terms of Ms, we quantitatively characterize the stochastic spiking coherence, and find that Ms reflects the degree of collective spiking coherence seen in the raster plot very well. Hence, the "statistical-mechanical" spike-based measure Ms may be used usefully to quantify the degree of stochastic spiking coherence in a statistical-mechanical way.

[1] W. Lim and S.-Y. Kim, "Statisitical-Mechanical Measure of Stochastic Spiking Coherence in A Population of Inhibitory Subthreshold Neuron", J. Comput. Neurosci. 31, 667-677 (2011).
[2] S.-Y. Kim and W. Lim, "Realistic thermodynamic and statistical-mechanical measures for neural synchronization," J. Neurosci. Methods 226, 161-170 (2014).

(2) Statistical-Mechanical Measure of Burst and Spike Synchronizations of Bursting Neurons

 We are interested in characterization of population synchronization of bursting neurons which exhibit both the slow bursting and the fast spiking timescales, in contrast to spiking neurons. Population synchronization may be well visualized in the raster plot of neural spikes which can be obtained in experiments. The instantaneous population firing rate (IPFR), which may be directly obtained from the raster plot of spikes, is often used as a realistic collective quantity describing population behaviors in both the computational and the experimental neuroscience. For the case of spiking neurons, realistic thermodynamic order parameter and statistical–mechanical spiking measure, based on IPFR, were introduced in our recent work to make practical characterization of spike synchronization. Here, we separate the slow bursting and the fast spiking timescales via frequency filtering, and extend the thermodynamic order parameter and the statistical–mechanical measure to the case of bursting neurons. Consequently, it is shown in explicit examples that both the order parameters and the statistical–mechanical measures may be effectively used to characterize the burst and spike synchronizations of bursting neurons.

[1] S.-Y. Kim and W. Lim, "Thermodynamic order parameters and statistical-mechanical measures for characterization of the burst and spike synchronizations of bursting neurons," Physica A 438, 544-559 (2015).
[2] S.-Y. Kim and W. Lim, "Frequency-domain order parameters for the burst and spike synchronization transitions of bursting neurons," Cognitive Neurodynamics 9, 411-421 (2015).


Stochastic Spiking Coherence in Networks of Subthreshold Neurons

 In recent years, much attention has been paid to brain rhythms. Synchronization of the firing activity in groups of neurons may be used for efficient sensory processing (e.g., visual binding). In addition to a constructive role of encoding sensory stimuli, neural synchronization is also correlated with pathological rhythms associated with neural diseases (e.g., epileptic seizures and tremors in Parkinson's disease). Many studies on collective dynamical behaviors in neural systems were made to understand the mechanisms of such synchronized firings. However, most of them were restricted to the suprathreshold case consisting of spontaneously firing (i.e., self-oscillating) neurons. We note that in addition to suprathreshold neurons, subthreshold neurons (which can fire only with the help of noise) also exist in real neural systems. Hence, we must take into consideration the existence of subthreshold neurons for the study on the population neurodynamics. For the subthreshold case, neurons cannot fire spontaneously without noise. Noise is usually considered as a nuisance, degrading the performance of dynamical systems. However, in certain circumstances, noise plays a constructive role in the emergence of dynamical order. A main subject of our study is to investigate stochastic spiking coherence (i.e., collective coherence between noise-induced firings) for the subthreshold case. Based on this study for the subthreshold case, we also study coherent population dynamics in heterogeneous ensembles composed of subthreshold and suprathreshold neurons.

(1) Stochastic Spiking Coherence in Coupled Excitatory Neurons

A. Characterization of Stochastic Spiking Coherence

We consider a large population of subthreshold Morris-Lecar neurons interacting via global instantaneous pulse-type excitatory synaptic coupling. By varying the noise intensity, we investigate numerically stochastic spiking coherence (i.e., collective coherence between noise-induced neural spikings). As the noise amplitude passes a threshold, a transition from an incoherent to a coherent state occurs. This coherent transition is described in terms of the "thermodynamic" order parameter, which concerns a macroscopic time-averaged fluctuation of the global potential. We note that such stochastic spiking coherence may be well visualized in terms of the raster plot of neural spikings (i.e., spatiotemporal plot of neural spikings), which is directly obtained in experiments. To quantitatively measure the degree of stochastic spiking coherence (seen in the raster plot), we introduce a new type of ``spiking coherence measure,'' by taking into consideration the average contribution of (microscopic) local neural spikings to the (macroscopic) global membrane potential. Hence, the spiking coherence measure may be regarded as a "statistical-mechanical" measure. Through competition between the constructive and the destructive roles of noise, stochastic spiking coherence is found to occur over a large range of intermediate noise intensities and to be well characterized in terms of the mutually complementary quantities of the order parameter and the spiking measure. Particularly, the spiking measure reflects the degree of stochastic spiking coherence seen in the raster plot very well.

[1] W. Lim and S.-Y. Kim, "Characterization of Stochastic Spiking Coherence in Coupled Neurons," J. Korean Phys. Soc. 51, 1427-1431 (2007).
[2] W. Lim and S.-Y. Kim, "Stochastic spiking coherence in coupled subthreshold Morris-Lecar neurons," Int. J. Mod. Phys. B 23, 703-710 (2009).

B. Stochastic Oscillator Death

 We consider an ensemble of subthreshold Morris-Lecar neurons interacting via global instantaneous pulse-type excitatory synaptic coupling. As the coupling strength passes a lower threshold, the coupling stimulates coherence between noise-induced spikings. This coherent transition is well described in terms of an order parameter. However, for sufficiently large coupling strength, "stochastic oscillator death" (i.e., quenching of noise-induced spikings), leading to breakup of collective spiking coherence, is found to occur. Using the techniques of nonlinear dynamics, we investigate the dynamical origin of stochastic oscillator death. Thus, we show that stochastic oscillator death occurs because each local neuron is attracted to a noisy equilibrium state via an infinite-period bifurcation. Furthermore, we introduce a new "statistical-mechanical" parameter, called the average firing probability, and quantitatively characterize a transition from firing to non-firing states which results from stochastic oscillator death. For a firing (non-firing) state, the averaging firing probability tends to be non-zero (zero) in the thermodynamic limit. We note that the role of the average firing probability for the firing-nonfiring transition is similar to that of the order parameter used for the coherence-incoherence transition.

[1] W. Lim and S.-Y. Kim, "Stochastic Oscillator Death in Globally Coupled Neural Systems," J. Korean Phys. Soc. 52, 1913-1917 (2008).
[2] W. Lim and S.-Y. Kim, "Coupling-induced spiking coherence in coupled subthreshold neurons," Int. J. Mod. Phys. B 23, 2149-2157 (2009).

(2) Stochastic Spiking Coherence in Coupled Inhibitory Neurons

Depending on the type of synaptic receptors (e.g., AMPA and GABA), the synaptic coupling may be excitatory or inhibitory. About 20 % of neurons are inhibitory ones in the human brain. For example, the principal neurons in the cortex are excitatory ones, while the interneurons are inhibitory ones. Through the efficient roles of the inhibitory neurons, the functions of the excitatory neurons are diversified and their computational abilities are much enhanced. In such a way, the brain functions may be well performed via balance of excitation and inhibition. Here, we study coherent population dynamics in coupled inhibitory neurons, and compare them with those in coupled excitatory neurons.

A. Statistical-Mechanical Measure of Stochastic Spiking Coherence in A Population of Inhibitory Subthreshold Neurons

 By varying the noise intensity, we study stochastic spiking coherence (i.e., collective coherence between noise-induced neural spikings) in an inhibitory population of subthreshold neurons (which cannot fire spontaneously without noise). This stochastic spiking coherence may be well visualized in the raster plot of neural spikes. For a coherent case, partially-occupied "stripes" (composed of spikes and indicating collective coherence) are formed in the raster plot. This partial occupation occurs due to "stochastic spike skipping" which is well shown in the multi-peaked interspike interval histogram. The main purpose of our work is to quantitatively measure the degree of stochastic spiking coherence seen in the raster plot. We introduce a new spike-based coherence measure Ms by considering the occupation pattern and the pacing pattern of spikes in the stripes. In particular, the pacing degree between spikes is determined in a statistical-mechanical way by quantifying the average contribution of (microscopic) individual spikes to the (macroscopic) ensemble-averaged global potential. This "statistical-mechanical" measure Ms is in contrast to the conventional measures such as the "thermodynamic" order parameter (which concerns the time-averaged fluctuations of the macroscopic global potential), the "microscopic" correlation-based measure (based on the cross-correlation between the microscopic individual potentials), and the measures of precise spike timing (based on the peri-stimulus time histogram). In terms of Ms, we quantitatively characterize the stochastic spiking coherence, and find that Ms reflects the degree of collective spiking coherence seen in the raster plot very well. Hence, the "statistical-mechanical" spike-based measure Ms may be used usefully to quantify the degree of stochastic spiking coherence in a statistical-mechanical way.

[1] W. Lim and S.-Y. Kim, "Statistical-Mechanical Measure of Stochastic Spiking Coherence in A Population of Inhibitory Subthreshold Neuron", J. Comput. Neurosci. 31, 667-677 (2011).

B. Inhibitory Coherence in A Heterogeneous Population of Subthreshold and Suprathreshold Type-I Neurons

We study inhibitory coherence (i.e., collective coherence by synaptic inhibition) in a population of globally coupled type-I neurons which can fire at arbitrarily low frequency. No inhibitory coherence is observed in a homogeneous population composed of only subthreshold neurons which exhibit noise-induced firings. In addition to subthreshold neurons, there exist spontaneously firing suprathreshold neurons in a noisy environment of a real brain. To take into consideration the effect of suprathreshold neurons on inhibitory coherence, we consider a heterogeneous population of subthreshold and suprathreshold neurons, and investigate the inhibitory coherence by increasing the fraction of suprathreshold neurons Psupra. As Psupra passes a threshold Psupra, suprathreshold neurons begin to synchronize and play the role of coherent inhibitors for the emergence of inhibitory coherence. Thus, regularly-oscillating population-averaged global potential appears for Psupra> P*supra. For this coherent case suprathreshold neurons exhibit sparse spike synchronization (i.e., individual potentials of suprathreshold neurons consist of coherent sparse spikings and coherent subthreshold small-amplitude hoppings). By virtue of their coherent inhibition, sparsely synchronized suprathreshold neurons suppress noisy activity of subthreshold neurons. Thus, subthreshold neurons exhibit hopping synchronization (i.e., only coherent subthreshold hopping oscillations without spikings appear in the individual potentials of subthreshold neurons). We also characterize the inhibitory coherence in terms of the "statistical-mechanical" spike-based and correlation-based measures which quantify the average contributions of the microscopic individual spikes and individual potentials to the macroscopic global potential. Finally, effect of sparse randomness of synaptic connectivity on the inhibitory coherence is briefly discussed.

[1] S.-Y. Kim, D.-G. Hong, J. Kim, and W. Lim, "Inhibitory coherence in a heterogeneous population of subthreshold and suprathreshold type-I neurons", J. Phys. A 45, 155102 (2012).

C. Effect of Sparse Random Connectivity on the Stochastic Spiking Coherence of Inhibitory Subthreshold Neurons

 We study the effect of network structure on the stochastic spiking coherence (i.e., collective coherence between noise-induced neural spikings) in an inhibitory population of subthreshold neurons (which cannot fire spontaneously without noise). Previously, stochastic spiking coherence was found to occur for the case of global coupling. However, "sparseness" of a real neural network is well known. Hence, we investigate the effect of sparse random connectivity on the stochastic spiking coherence by varying the average number of synaptic inputs per neuron Msyn. It is thus found that stochastic spiking coherence emerges if Msyn is larger than a (very small) critical value M*syn, independently of the network size N. This stochastic spiking coherence may be well visualized in the raster plot of neural spikes. For a coherent case, partially-occupied stripes appear. As Msyn is decreased from N-1 (globally-coupled case), the average occupation degree of spikes per stripe increases very slowly. On the other hand, the average pacing degree between spikes per stripe decreases slowly, but near M*syn its decrease becomes very rapid. This decrease in the pacing degree can also be well seen through merging of multiple peaks in the interspike interval histograms. Due to the effect of the pacing degree, the degree of stochastic spiking coherence decreases abruptly near the threshold M*syn.

[1] D.-G. Hong, S.-Y. Kim, and W. Lim, "Effect of Sparse Random Connectivity on the Stochastic Spiking Coherence of Inhibitory Subthreshold Neurons," J. Korean Phys. Soc. 59, 2840-2846 (2011).

(3) Stochastic Spiking Coherence in Two Populations of Excitatory and Inhibitory Neurons

Human brain exhibits their functions well through balance between the excitatory and the inhibitory neurons. The ratio of inhibitory neurons in the brain is about 20%. We are interested in collective spiking coherence in two populations of excitatory and inhibitory neurons. Particularly, we study the effect of the interpopulation coupling on such spiking coherence. So, by varying the interpopulation coupling strength to keep the E/I balance (i.e., the ratio of AMPA and the GABA conductances is equal for both E- and I-neurons), we investigate coherent population dynamics in each population. Such population dynamics is analyzed using the techniques of the nonlinear dynamics and the statistical mechanics. We also introduce the interpopulation spiking measure between the source and the target populations, and characterize the degree of the interpopulation effect.

A. Interpopulation Effect on Stochastic Spiking Coherence in Two Populations of Excitatory and Inhibitory Subthreshold Type-I Neurons

 We consider two populations of excitatory and inhibitory subthreshold type-I neurons which can fire at arbitrary low frequency. No coherence occurs in the single inhibitory population, while stochastic excitatory coherence appears in the single excitatory population. Through the coupling between the excitatory source population and the inhibitory target population, coherence may appear in the inhibitory population. By varying the interpopulation coupling strength to keep the E/I balance, we investigate the effect of the interpopulation coupling on the stochastic spiking coherence in each population. The population dynamics in each population is analyzed using the techniques of the nonlinear dynamics and the statistical mechanics. Particularly, we introduce the interpopulation spiking measure between the source and the target population, and characterize the degree of the interpopulation-coupling effect.  


Strange Nonchaotic Firing in the Quasiperiodically-forced Neuron

 We study the transition from a silent state to a spiking state by varying the DC stimulus in the quasiperiodically-forced neuron. For this quasiperiodically-forced case, a new type of strange nonchaotic (SN) firing (spiking or bursting) state is found to appear between the silent state and the chaotic firing state as an intermediate one. Using a rational approximation to the quasiperiodic forcing, we investigate the mechanism for the appearance of such an SN firing state. We thus find that a smooth torus (corresponding to the silent state) is transformed into an SN firing attractor via a phase-dependent bifurcation. This is in contrast to the periodically-forced case where the silent state transforms directly to a chaotic spiking state. These SN firing states are also found to be be aperiodic complex ones, as in the case of chaotic spiking states. Hence, aperiodic complex spikings may result from two dynamically different states with strange geometry (one is chaotic and the other one is nonchaotic).

[1] W. Lim and S.-Y. Kim, "Strange nonchaotic bursting in the quasiperiodically forced Hindmarsh-Rose neuron", J. Korean Phys. Soc. 57, 1356~1362 (2010).
[2] W. Lim and S.-Y. Kim "Strange nonchaotic spiking in the quasiperiodically-forced Hodgkin-Huxley neuron," J. Korean Phys. Soc. 57, 23-29 (2010).


Strange Nonchaotic Response in the Quasiperiodically-forced Neuron

 We study dynamical responses of the self-oscillating neuron under quasiperiodic stimulation. For the case of periodic stimulation on the self-oscillating neuron, a transition from a periodic to a chaotic oscillation occurs through period doublings. We investigate the effect of the quasiperiodic forcing on this period-doubling route to chaotic oscillation. In contrast to the periodically-forced case, a new type of strange nonchaotic (SN) oscillating states (that are geometrically strange but have no positive Lyapunov exponents) is thus found to appear between the regular and chaotic oscillating states. Strange fractal geometry of these SN oscillating states, which is characterized in terms of the phase sensitivity exponent and the distribution of local finite-time Lyapunov exponent, leads to aperiodic complex spikings. Diverse routes to SN oscillations are found, as in the quasiperiodically forced logistic map.

[1] W. Lim and S.-Y. Kim, "Strange nonchaotic oscillations in the quasiperiodically forced Hodgkin-Huxley neuron," J. Phys. A 42, 265103 (2009). [PDF]
[2] W. Lim, S.-Y. Kim, and Y. Kim, "Strange nonchaotic responses of the quasiperiodically forced Morris-Lecar neuron," Prog. Theor. Phys. 121, 671-683 (2009). [PDF]

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