Correction to: Population and individual firing behaviors in sparsely synchronized rhythms in the hippocampal dentate gyrus.

[This corrects the article DOI: 10.1007/s11571-021-09728-4.].

Disynaptic effect of hilar cells on pattern separation in a spiking neural network of hippocampal dentate gyrus.

We study the disynaptic effect of the hilar cells on pattern separation in a spiking neural network of the hippocampal dentate gyrus (DG). The principal granule cells (GCs) in the DG perform pattern separation, transforming similar input patterns into less-similar output patterns. In our DG network, the hilus consists of excitatory mossy cells (MCs) and inhibitory HIPP (hilar perforant path-associated) cells. Here, we consider the disynaptic effects of the MCs and the HIPP cells on the GCs, mediated by the inhibitory basket cells (BCs) in the granular layer; MC → BC → GC and HIPP → BC → GC. The MCs provide disynaptic inhibitory input (mediated by the intermediate BCs) to the GCs, which decreases the firing activity of the GCs. On the other hand, the HIPP cells disinhibit the intermediate BCs, which leads to increasing the firing activity of the GCs. In this way, the disynaptic effects of the MCs and the HIPP cells are opposite. We investigate change in the pattern separation efficacy by varying the synaptic strength K ( BC , X ) [from the pre-synaptic X (= MC or HIPP) to the post-synaptic BC]. Thus, sparsity for the firing activity of the GCs is found to improve the efficacy of pattern separation, and hence the disynaptic effects of the MCs and the HIPP cells on the pattern separation become opposite ones. In the combined case when simultaneously changing both K ( BC , MC ) and K ( BC , HIPP ) , as a result of balance between the two competing disynaptic effects of the MCs and the HIPP cells, the efficacy of pattern separation is found to become the highest at their original default values where the activation degree of the GCs is the lowest. We also note that, while the GCs perform pattern separation, sparsely synchronized rhythm is found to appear in the population of the GCs. Hence, we examine quantitative association between population and individual firing behaviors in the sparsely synchronized rhythm and pattern separation. They are found to be strongly correlated. Consequently, the better the population and individual firing behaviors in the sparsely synchronized rhythm are, the more pattern separation efficacy becomes enhanced.

Population and individual firing behaviors in sparsely synchronized rhythms in the hippocampal dentate gyrus.

We investigate population and individual firing behaviors in sparsely synchronized rhythms (SSRs) in a spiking neural network of the hippocampal dentate gyrus (DG). The main encoding granule cells (GCs) are grouped into lamellar clusters. In each GC cluster, there is one inhibitory (I) basket cell (BC) along with excitatory (E) GCs, and they form the E-I loop. Winner-take-all competition, leading to sparse activation of the GCs, occurs in each GC cluster. Such sparsity has been thought to enhance pattern separation performed in the DG. During the winner-take-all competition, SSRs are found to appear in each population of the GCs and the BCs through interaction of excitation of the GCs with inhibition of the BCs. Sparsely synchronized spiking stripes appear successively with the population frequency f p ( = 13.1 Hz) in the raster plots of spikes. We also note that excitatory hilar mossy cells (MCs) control the firing activity of the GC-BC loop by providing excitation to both the GCs and the BCs. SSR also appears in the population of MCs via interaction with the GCs (i.e., GC-MC loop). Population behaviors in the SSRs are quantitatively characterized in terms of the synchronization measures. In addition, we investigate individual firing activity of GCs, BCs, and MCs in the SSRs. Individual GCs exhibit random spike skipping, leading to a multi-peaked inter-spike-interval histogram, which is well characterized in terms of the random phase-locking degree. In this case, population-averaged mean-firing-rate (MFR) < f i ( GC ) > is less than the population frequency f p . On the other hand, both BCs and MCs show "intrastripe" burstings within stripes, together with random spike skipping. Thus, the population-averaged MFR ⟨ f i ( X ) ⟩ ( X = MC and BC) is larger than f p , in contrast to the case of the GCs. MC loss may occur during epileptogenesis. With decreasing the fraction of the MCs, changes in the population and individual firings in the SSRs are also studied. Finally, quantitative association between the population/individual firing behaviors in the SSRs and the winner-take-all competition is discussed.

Dynamical origin for winner-take-all competition in a biological network of the hippocampal dentate gyrus.

We consider a biological network of the hippocampal dentate gyrus (DG). Computational models suggest that the DG would be a preprocessor for pattern separation (i.e., a process transforming a set of similar input patterns into distinct nonoverlapping output patterns) which could facilitate pattern storage and retrieval in the CA3 area of the hippocampus. The main encoding cells in the DG are the granule cells (GCs) which receive the input from the entorhinal cortex (EC) and send their output to the CA3. We note that the activation degree of GCs is very low (∼5%). This sparsity has been thought to enhance the pattern separation. We investigate the dynamical origin for winner-take-all (WTA) competition which leads to sparse activation of the GCs. The whole GCs are grouped into lamellar clusters. In each cluster, there is one inhibitory (I) basket cell (BC) along with excitatory (E) GCs. There are three kinds of external inputs into the GCs: the direct excitatory EC input; the indirect feedforward inhibitory EC input, mediated by the HIPP (hilar perforant path-associated) cells; and the excitatory input from the hilar mossy cells (MCs). The firing activities of the GCs are determined via competition between the external E and I inputs. The E-I conductance ratio R_{E-I}^{(con)}^{*} (given by the time average of the ratio of the external E to I conductances) may represent well the degree of such external E-I input competition. It is thus found that GCs become active when their R_{E-I}^{(con)}^{*} is larger than a threshold R_{th}^{*}, and then the mean firing rates of the active GCs are strongly correlated with R_{E-I}^{(con)}^{*}. In each cluster, the feedback inhibition from the BC may select the winner GCs. GCs with larger R_{E-I}^{(con)}^{*} than the threshold R_{th}^{*} survive, and they become winners; all the other GCs with smaller R_{E-I}^{(con)}^{*} become silent. In this way, WTA competition occurs via competition between the firing activity of the GCs and the feedback inhibition from the BC in each cluster. Finally, we also study the effects of MC death and adult-born immature GCs on the WTA competition.

Influence of various temporal recoding on pavlovian eyeblink conditioning in the cerebellum.

We consider the Pavlovian eyeblink conditioning (EBC) via repeated presentation of paired conditioned stimulus (tone) and unconditioned stimulus (US; airpuff). In an effective cerebellar ring network, we change the connection probability p c from Golgi to granule (GR) cells, and make a dynamical classification of various firing patterns of the GR cells. Individual GR cells are thus found to show various well- and ill-matched firing patterns relative to the US timing signal. Then, these variously-recoded signals are fed into the Purkinje cells (PCs) through the parallel-fibers (PFs). Based on such unique dynamical classification of various firing patterns, we make intensive investigations on the influence of various temporal recoding (i.e., firing patterns) of the GR cells on the synaptic plasticity of the PF-PC synapses and the subsequent learning process for the EBC. We first note that the variously-recoded PF signals are effectively depressed by the (error-teaching) instructor climbing-fiber (CF) signals from the inferior olive neuron. In the case of well-matched PF signals, they are strongly depressed through strong long-term depression (LTD) by the instructor CF signals due to good association between the in-phase PF and the instructor CF signals. On the other hand, practically no LTD occurs for the ill-matched PF signals because most of them have no association with the instructor CF signals. This kind of "effective" depression at the PF-PC synapses coordinates firings of PCs effectively, which then makes effective inhibitory coordination on the cerebellar nucleus neuron [which elicits conditioned response (CR; eyeblink)]. When the learning trial passes a threshold, acquisition of CR begins. In this case, the timing degree T d 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 number of trials, 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 T d 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 number of learning trials, the (overall) learning efficiency degree L e (taking into consideration both timing and strength of CR) for the CR is increased, and eventually it becomes saturated. Finally, we also discuss dependence of the variety degree for firing patterns of the GR cells and the saturated learning efficiency degree L e of the CR on p c and their relations.

Effect of diverse recoding of granule cells on optokinetic response in a cerebellar ring network with synaptic plasticity.

We consider a cerebellar ring network for the optokinetic response (OKR), and investigate the effect of diverse recoding of granule (GR) cells on OKR by varying the connection probability pc from Golgi to GR cells. For an optimal value of pc∗(=0.06), individual GR cells exhibit diverse spiking patterns which are in-phase, anti-phase, or complex out-of-phase with respect to their population-averaged firing activity. Then, these diversely-recoded signals via parallel fibers (PFs) from GR cells are effectively depressed by the error-teaching signals via climbing fibers from the inferior olive which are also in-phase ones. 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. This kind of "effective" depression (i.e., strong/weak LTD) at the PF-PC synapses causes a big modulation in firings of PCs, which then exert effective inhibitory coordination on the vestibular nucleus (VN) neuron (which evokes 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 at about the 300th cycle. By varying pc from pc∗, we find that a plot of saturated learning gain degree Lg∗ versus pc forms a bell-shaped curve with a peak at pc∗ (where the diversity degree in spiking patterns of GR cells is also maximum). Consequently, the more diverse in recoding of GR cells, the more effective in motor learning for the OKR adaptation.

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.

Cluster burst synchronization in a scale-free network of inhibitory bursting neurons.

We consider a scale-free network of inhibitory Hindmarsh-Rose (HR) bursting neurons, and make a computational study on coupling-induced cluster burst synchronization by varying the average coupling strength J 0 . For sufficiently small J 0 , non-cluster desynchronized states exist. However, when passing a critical point J c ∗ ( ≃ 0.16 ) , the whole population is segregated into 3 clusters via a constructive role of synaptic inhibition to stimulate dynamical clustering between individual burstings, and thus 3-cluster desynchronized states appear. As J 0 is further increased and passes a lower threshold J l ∗ ( ≃ 0.78 ) , a transition to 3-cluster burst synchronization occurs due to another constructive role of synaptic inhibition to favor population synchronization. In this case, HR neurons in each cluster make burstings every 3rd cycle of the instantaneous burst rate R w ( t ) of the whole population, and exhibit burst synchronization. However, as J 0 passes an intermediate threshold J m ∗ ( ≃ 5.2 ) , HR neurons fire burstings intermittently at a 4th cycle of R w ( t ) via burst skipping rather than at its 3rd cycle, and hence they begin to make intermittent hoppings between the 3 clusters. Due to such intermittent intercluster hoppings via burst skippings, the 3 clusters become broken up (i.e., the 3 clusters are integrated into a single one). However, in spite of such break-up (i.e., disappearance) of the 3-cluster states, (non-cluster) burst synchronization persists in the whole population, which is well visualized in the raster plot of burst onset times where bursting stripes (composed of burst onset times and indicating burst synchronization) appear successively. With further increase in J 0 , intercluster hoppings are intensified, and bursting stripes also become dispersed more and more due to a destructive role of synaptic inhibition to spoil the burst synchronization. Eventually, when passing a higher threshold J h ∗ ( ≃ 17.8 ) a transition to desynchronization occurs via complete overlap between the bursting stripes. Finally, we also investigate the effects of stochastic noise on both 3-cluster burst synchronization and intercluster hoppings.

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 Barabási-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 Δ l in the SFN.

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 synchronization (with higher spiking measure) is found to get better via LTD, while bad synchronization (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.

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). However, STDP was not considered in previous works on stochastic burst synchronization (SBS) between noise-induced burstings of sub-threshold neurons. Here, we study the effect of additive STDP on SBS by varying the noise intensity D in the Barabási-Albert scale-free network (SFN). One of our main findings is a Matthew effect in synaptic plasticity which occurs due to a positive feedback process. Good burst synchronization (with higher bursting measure) gets better via long-term potentiation (LTP) of synaptic strengths, while bad burst synchronization (with lower bursting measure) 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. We also investigate the effects of network architecture on SBS by varying the symmetric attachment degree [Formula: see text] and the asymmetry parameter [Formula: see text] in the SFN, and Matthew effects are also found to occur by varying [Formula: see text] and [Formula: see text]. Furthermore, emergences of LTP and LTD of synaptic strengths are investigated in details via our own microscopic methods 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 instantaneous individual burst rates (IIBRs). Finally, a multiplicative STDP case (depending on states) with soft bounds is also investigated in comparison with the additive STDP case (independent of states) with hard bounds. Due to the soft bounds, a Matthew effect with some quantitative differences is also found to occur for the case of multiplicative STDP.

Stochastic spike synchronization in a small-world neural network with spike-timing-dependent plasticity.

We consider the Watts-Strogatz small-world network (SWN) consisting of subthreshold neurons which exhibit noise-induced spikings. This neuronal network has adaptive dynamic synaptic strengths governed by the spike-timing-dependent plasticity (STDP). In previous works without STDP, stochastic spike synchronization (SSS) between noise-induced spikings of subthreshold neurons was found to occur in a range of intermediate noise intensities. Here, we investigate the effect of additive STDP on the SSS 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 of synaptic strengths, while bad synchronization gets worse via long-term depression. Emergences of long-term potentiation and long-term depression of synaptic strengths are intensively investigated via microscopic studies based on the pair-correlations between the pre- and the post-synaptic IISRs (instantaneous individual spike rates) as well as the distributions of time delays between the pre- and the post-synaptic spike times. Furthermore, the effects of multiplicative STDP (which depends on states) on the SSS are studied and discussed in comparison with the case of additive STDP (independent of states). These effects of STDP on the SSS in the SWN are also compared with those in the regular lattice and the random graph.

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

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 [Formula: see text] (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 [Formula: see text]. With increasing A, based on [Formula: see text], we discuss the cross-correlations between the two sub-populations, affecting the dynamical responses to S(t).

Emergence of ultrafast 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, and investigate the effect of network architecture on emergence of synchronized brain rhythms by varying the fraction of LR interneurons plong. The betweenness centralities of the LR and SR interneurons (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 the same betweenness centralities. For small plong, the load of communication traffic is much concentrated on a few LR interneurons. However, as plong is increased, the number of LR connections (coming from LR interneurons) increases, and then the load of communication traffic is less concentrated on LR interneurons, which leads to better efficiency of global communication between interneurons. Sparsely synchronized rhythms are thus found to emerge when passing a small critical value plong(c)(≃0.16). The population frequency of the sparsely synchronized rhythm is ultrafast (higher than 100 Hz), while the mean firing rate of individual interneurons is much lower (∼30 Hz) due to stochastic and intermittent neural discharges. These dynamical behaviors in the inhomogeneous SWN are also compared with those in the homogeneous Watts-Strogatz SWN, in connection with their network topologies. Particularly, we note that the main difference between the two types of SWNs lies in the distribution of betweenness centralities. Unlike the case of the Watts-Strogatz SWN, dynamical responses to external stimuli vary depending on the type of stimulated interneurons in the inhomogeneous SWN. We consider two cases of external time-periodic stimuli applied to sub-populations of the 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.

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 α- and ß-processes. The α-process corresponds to a directed version of the Barabási-Albert SFN model with growth and preferential attachment, while for the ß-process only preferential attachments between pre-existing nodes are made without addition of new nodes. We first consider the "pure" α-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 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 α-process, and (3) the occurrence probability of the ß-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 just taking into consideration Lp and Bc (affecting global communication between nodes) is not sufficient to understand emergence of population synchronization in SFNs, but in addition to them, the in-degree distribution (affecting individual dynamics) must also be considered to fully understand for the effective population synchronization.

Effect of intermodular connection on fast sparse synchronization in clustered small-world neural networks.

We consider a clustered network with small-world subnetworks of inhibitory fast spiking interneurons and investigate the effect of intermodular connection on the emergence of fast sparsely synchronized rhythms by varying both the intermodular coupling strength J(inter) and the average number of intermodular links per interneuron M(syn)(inter). In contrast to the case of nonclustered 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 intramodular dynamics of subnetworks 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 intramodular dynamics of subnetworks make perfect matching. We introduce a realistic cross-correlation modularity measure, representing the matching degree between the instantaneous subpopulation spike rates of the subnetworks, and examine whether the sparse synchronization is global or modular. Depending on its magnitude, the intermodular coupling strength J(inter) seems to play "dual" roles for the pacing between spikes in each subnetwork. For large J(inter), due to strong inhibition it plays a destructive role to "spoil" the pacing between spikes, while for small J(inter) it plays a constructive role to "favor" the pacing between spikes. Through competition between the constructive and the destructive roles of J(inter), there exists an intermediate optimal J(inter) at which the pacing degree between spikes becomes maximal. In contrast, the average number of intermodular links per interneuron M(syn)(inter) seems to play a role just to favor the pacing between spikes. With increasing M(syn)(inter), 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.

Fast sparsely synchronized brain rhythms in a scale-free neural network.

We consider a directed version of the Barabási-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

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) [Formula: see text], 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 [Formula: see text], was introduced in our recent work to characterize the spike synchronization transition. Unlike the case of spiking neurons, the IPFR [Formula: see text] of bursting neurons exhibits population behaviors with both the slow bursting and the fast spiking timescales. For our aim, we decompose the IPFR [Formula: see text] into the instantaneous population bursting rate [Formula: see text] (describing the bursting behavior) and the instantaneous population spike rate [Formula: see text] (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" [Formula: see text] and [Formula: see text] of the bursting and spiking peaks in the power spectral densities of [Formula: see text] and [Formula: see text] (i.e., "signal to noise" ratio of the spectral peak height and its relative width). Through calculation of [Formula: see text] and [Formula: see text], 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.

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 [Formula: see text]-[Formula: see text] plane ([Formula: see text]: synaptic inhibition strength and [Formula: see text]: noise intensity). As the rewiring probability [Formula: see text] is decreased from 1 (random network) to 0 (regular lattice), the region of spike synchronization shrinks rapidly in the [Formula: see text]-[Formula: see text] 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.

Realistic thermodynamic and statistical-mechanical measures for neural synchronization.

Synchronized brain rhythms, associated with diverse cognitive functions, have been observed in electrical recordings of brain activity. Neural synchronization may be well described by using the population-averaged global potential VG in computational neuroscience. The time-averaged fluctuation of VG plays the role of a "thermodynamic" order parameter O used for describing the synchrony-asynchrony transition in neural systems. Population spike synchronization may be well visualized in the raster plot of neural spikes. The degree of neural synchronization seen in the raster plot is well measured in terms of a "statistical-mechanical" spike-based measure Ms introduced by considering the occupation and the pacing patterns of spikes. The global potential VG is also used to give a reference global cycle for the calculation of Ms. Hence, VG becomes an important collective quantity because it is associated with calculation of both O and Ms. However, it is practically difficult to directly get VG in real experiments. To overcome this difficulty, instead of VG, we employ the instantaneous population spike rate (IPSR) which can be obtained in experiments, and develop realistic thermodynamic and statistical-mechanical measures, based on IPSR, to make practical characterization of the neural synchronization in both computational and experimental neuroscience. Particularly, more accurate characterization of weak sparse spike synchronization can be achieved in terms of realistic statistical-mechanical IPSR-based measure, in comparison with the conventional measure based on VG.