Abstract
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.
I. INTRODUCTION
In recent years, brain rhythms have attracted much attention [1–15]. Particularly, we are interested in fast sparsely synchronized rhythms, associated with diverse cognitive functions (e.g., sensory perception, feature integration, selective attention, and memory formation) [16]. At the population level, fast sparsely synchronous oscillations [e.g., gamma rhythm (30–100 Hz) during awake behaving states and rapid eye movement sleep and sharp-wave ripple (100–200 Hz) during quiet sleep and awake immobility] have been observed in local field potential recordings, while at the cellular level individual neuronal recordings have been found to exhibit irregular and intermittent spike discharges like Geiger counters [17–23]. Thus, single-cell firing activity differs distinctly from the population oscillatory behavior. We note that these fast sparsely synchronized rhythms are in contrast to fully synchronized rhythms where individual neurons fire regularly at the population frequency like clocks. Under the balance between strong external noise and strong recurrent inhibition, fast sparse synchronization (FSS) was found to appear in both random networks [24–27] and globally-coupled networks [28, 29]. In brain networks, architecture of synaptic connections has been found to have complex topology (e.g., small-worldness and scale-freeness) which is neither regular nor completely random [30–37]. In our recent works [38, 39], as complex networks we employed the small-world network and the scale-free network, and studied the effects of network architecture on emergence of FSS.
In the previous works on FSS, synaptic coupling strengths were static. However, in real brains synaptic strengths may change to adapt to the environment (i.e., they can be potentiated [40–42] or depressed [43–46]). These adjustments of synapses are called the synaptic plasticity which provides the basis for learning, memory, and development [47]. Regarding the synaptic plasticity, we consider a spike-timing-dependent plasticity (STDP) [48–55]. For the STDP, the synaptic strengths vary via an update rule depending on the relative time difference between the pre- and the post-synaptic spike times. Many models for STDP have been employed to explain results on synaptic modifications occurring in diverse neuroscience topics for health and disease (e.g., temporal sequence learning [56], temporal pattern recognition [57], coincidence detection [58], navigation [59], direction selectivity [60], memory consolidation [61], competitive/selective development [62], and deep brain stimulation [63]). Recently, the effects of STDP on population synchronization in ensembles of coupled neurons were also studied in various aspects [64–69].
A neural circuit in the brain cortex is composed of a few types of excitatory principal cells and diverse types of inhibitory interneurons. These interneurons make up about 20 percent of all cortical neurons, and exhibit diversity in their morphologies and functions [70]. By providing a synchronous oscillatory output to the principal cells, interneuronal networks play the role of backbones of many cortical rhythms [7, 71–73]. Synaptic plasticity of excitatory and inhibitory connections is of great interest because it controls the efficacy of potential computational functions of excitation and inhibition. Studies of synaptic plasticity have been mainly focused on synaptic connections between excitatory pyramidal cells, because excitatory-to-excitatory (E to E) connections are most prevalent in the cortex and they form a relatively homogeneous population [57, 58, 74–80]. An asymmetric Hebbian time window was employed for the excitatory STDP (eSTDP) update rule [48–55]. When a pre-synaptic spike precedes a post-synaptic spike, long-term potentiation (LTP) occurs; otherwise, long-term depression (LTD) appears. On the other hand, plasticity of inhibitory connections has attracted less attention mainly because of experimental obstacles and diversity of interneurons [81–85]. With the advent of fluorescent labeling and optical manipulation of neurons according to their genetic types [86, 87], inhibitory plasticity has also begun to be focused. Particularly, studies on iSTDP of inhibitory-toexcitatory (I to E) connections have been much made. Thus, inhibitory STDP (iSTDP) has been found to be diverse and cell-specific [81–85, 88–94].
In this paper, we consider an inhibitory Watts-Strogatz small-world network (SWN) of fast spiking (FS) interneurons [95–97], and investigate the effect of iSTDP [of inhibitory-to-inhibitory (I to I) connections] on FSS by varying the noise intensity D. We employ an asymmetric anti-Hebbian time window for the iSTDP update rule, in contrast to the Hebbian time window for the eSTDP [63, 66]. Then, strengths of synaptic inhibition {Jij} change with time, and eventually, they become saturated after a sufficiently long time. Depending on D, mean values of saturated synaptic inhibition strengths are potentiated [long-term potentiation (LTP)] or depressed [long-term depression (LTD)], when compared with the initial mean value of synaptic inhibition strengths. On the other hand, dispersions from the mean values of LTP/LTD are much increased in comparison with the initial dispersion, irrespectively of D.
For the case of iSTDP, both the mean value and the dispersion (for the distribution of synaptic inhibition strengths) affect population synchronization. The LTD (LTP) tends to increase (decrease) the degree of FSS due to decrease (increase) in the mean value of synaptic inhibition strengths, and the increased dispersions decrease the degree of FSS. For 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) gets better via LTD; in some other cases where dispersions are dominant, the degree of good synchronization may be decreased even in the case of LTD. On the other hand, in all cases bad synchronization (with lower spiking measure) gets worse via LTP. This kind of Matthew effect (valid in most cases of LTD) is in contrast to that in the case of eSTDP where good (bad) synchronization gets better (worse) via LTP (LTD) [68, 69]; the role of LTD (LTP) in the case of iSTDP is similar to that of LTP (LTD) for the case of eSTDP. Emergences of LTD and LTP of synaptic inhibition strengths are also investigated through a microscopic method based on the distributions of time delays between the nearest spiking times of the pre- and the post-synaptic interneurons. Moreover, we study the effects of network architecture on FSS by varying the rewiring probability p of the SWN in the presence of iSTDP.
This paper is organized as follows. In Sec. II, we describe an inhibitory Watts-Strogatz SWN of FS interneurons with inhibitory synaptic plasticity. Then, in Sec. III the effects of iSTDP on FSS are investigated. Finally, we give summary and discussion in Sec. IV.
II. WATTS-STROGATZ SWN OF FS IZHIKEVICH INTERNEURONS WITH INHIBITORY SYNAPTIC PLASTICITY
We consider an inhibitory directed Watts-Strogatz SWN, composed of N FS interneurons equidistantly placed on a one-dimensional ring of radius N/2π. The Watts-Strogatz SWN interpolates between a regular lattice with high clustering (corresponding to the case of p = 0) and a random graph with short average path length (corresponding to the case of p = 1) via random uniform rewiring with the probability p [95–97]. For p = 0, we start with a directed regular ring lattice with N nodes where each node is coupled to its first Msyn neighbors (Msyn/2 on either side) via outward synapses, and rewire each outward connection uniformly at random over the whole ring with the probability p (without self-connections and duplicate connections). This Watts-Strogatz SWN model may be regarded as a cluster-friendly extension of the random network by reconciling the six degrees of separation (small-worldness) [98, 99] with the circle of friends (clustering). These SWNs with predominantly local connections and rare long-range connections were employed in many recent works on various subjects of neurodynamics [100–111].
As elements in our SWN, we choose the Izhikevich inhibitory FS interneuron model which is not only biologically plausible, but also computationally efficient [112–115]. Unlike Hodgkin-Huxley-type conductance-based models, the Izhikevich model matches neuronal dynamics by tuning the parameters instead of matching neuronal electrophysiology. The parameters k and b are associated with the neuron’s rheobase and input resistance, and a, c, and d are the recovery time constant, the after-spike reset value of v, and the after-spike jump value of u, respectively. Tuning these parameters, the Izhikevich neuron model may produce 20 of the most prominent neuro-computational features of biological neurons [112–115]. Particularly, the Izhikevich model is used to reproduce the six most fundamental classes of firing patterns observed in the mammalian neocortex; (a) excitatory regular spiking pyramidal neurons, (b) inhibitory FS interneurons, (c) intrinsic bursting neurons, (d) chattering neurons, (e) low-threshold spiking neurons and (f) late spiking neurons [114]. Here, we use the parameter values for the FS interneurons (which do not fire postinhibitory rebound spikes) in the layer 5 rat visual cortex, which are listed in the 1st item of Table I (see the caption of Fig. 8.27 in [114]).
The following equations (1)–(6) govern the population dynamics in the SWN: with the auxiliary after-spike resetting: where
Here, the state of the ith neuron at a time t is characterized by two state variables: the membrane potential vi and the recovery current ui. In Eq. (1), C is the membrane capacitance, vr is the resting membrane potential, and vt is the instantaneous threshold potential. After the potential reaches its apex (i.e., spike cutoff value) vp, the membrane potential and the recovery variable are reset according to Eq. (3). The units of the capacitance C, the potential v, the current u and the time t are pF, mV, pA, and msec, respectively. The parameter values used in our computations are listed in Table I. More details on the random external excitatory input to each Izhikevich FS interneuron, the synaptic currents and plasticity, and the numerical method for integration of the governing equations are given in the following subsections.
A. Random External Excitatory Input to Each Izhikevich FS Interneuron
Each interneuron in the network receives stochastic external excitatory input Iext,i from other brain regions, not included in the network (i.e., corresponding to background excitatory input) [24–27, 71]. Then, Iext,i may be modeled in terms of its time-averaged constant Ii and an independent Gaussian white noise ξi (i.e., corresponding to fluctuation of Iext,i from its mean) [see the 3rd and the 4th terms in Eq. (1)] satisfying 〈ξi (t)〉 = 0 and 〈ξi (t) ξj (t′)〉 = δij δ(t − t′), where 〈· · ·〉 denotes the ensemble average. The intensity of the noise ξi is controlled by using the parameter D. In the absence of noise (i.e., D = 0), the Izhikevich interneuron exhibits a jump from a resting state to a spiking state via subcritical Hopf bifurcation for Ih = 73.7 by absorbing an unstable limit cycle born via a fold limit cycle bifurcation for Il = 72.8, as shown in Fig. 1(a) [38]. Hence, the Izhikevich FS interneuron shows type-II excitability because it begins to fire with a non-zero frequency [see Fig. 1(b)] [116, 117]. As I is increased from Ih, the mean firing rate (MFR) f increases monotonically, as shown in Fig. 1(c). Throughout this paper, we consider a suprathreshold case such that the value of Ii is chosen via uniform random sampling in the range of [680,720], as shown in the 2nd item of Table I; for the middle value of I = 700, the membrane potential v oscillates very fast with f = 271 Hz [see Fig. 1(d)].
B. Synaptic Currents and Plasticity
The last term in Eq. (1) represents the synaptic couplings of Izhikevich FS interneurons. Isyn,i of Eq. (5) represents a synaptic current injected into the ith interneuron, and Vsyn is the synaptic reversal potential. The synaptic connectivity is given by the connection weight matrix W (={wij}) where wij = 1 if the interneuron j is presynaptic to the interneuron i; otherwise, wij = 0. Here, the synaptic connection is modeled in terms of the Watts-Strogatz SWN. Then, the in-degree of the ith neuron, (i.e., the number of synaptic inputs to the interneuron i) is given by . For this case, the average number of synaptic inputs per neuron is given by . Throughout the paper, Msyn = 50 (see the 4th item of Table I). The fraction of open synaptic ion channels at time t is denoted by s (t). The time course of sj (t) of the jth neuron is given by a sum of delayed double-exponential functions [see Eq. (6)], where τl is the synaptic delay, and and Fj are the fth spike and the total number of spikes of the jth interneuron at time t, respectively. Here, E (t) [which corresponds to contribution of a presynaptic spike occurring at time 0 to sj (t) in the absence of synaptic delay] is controlled by the two synaptic time constants: synaptic rise time τr and decay time τd, and Θ(t) is the Heaviside step function: Θ(t) = 1 for t ≥ 0 and 0 for t < 0. For the inhibitory GABAergic synapse (involving the GABAA receptors), the values of τl, τr, τd, and Vsyn are listed in the 3rd item of Table I [26].
The coupling strength of the synapse from the jth pre-synaptic interneuron to the ith post-synaptic interneuron is Jij. Here, we consider a multiplicative iSTDP (depen-dent on states) for the synaptic strengths {Jij} [67, 118]. To avoid unbounded growth and elimination of synaptic connections, we set a range with the upper and the lower bounds: Jij ∈ [Jl, Jh], where Jl = 0.0001 and Jh = 2000. Initial synaptic strengths are normally distributed with the mean J0(= 700) and the standard deviation σ0 (= 5). With increasing time t, the synaptic strength for each synapse is updated with a nearest-spike pair-based STDP rule [119]: where δ (= 0.05) is the update rate, J* = Jh (Jl) for the LTP (LTD), and ΔJij (Δtij) is the synaptic modification depending on the relative time difference Δtij between the nearest spike times of the post-synaptic interneuron i and the pre-synaptic interneuron j. We use an asymmetric anti-Hebbian time window for the synaptic modification ΔJij (Δtij) [63, 66]: where A+ = 1.0, A− = 1.1, τ+ = 11.5 msec, and τ− = 12 msec (these values are also given in the 5th item of Table I). For the case of Δtij> 0, LTD occurs, while LTP takes place in the case of tij < 0, in contrast to the Hebbian time window for the eSTDP where LTP (LTD) occurs for Δtij> (<)0 [68].
C. Numerical Method for Integration
Numerical integration of stochastic differential Eqs. (1)–(6) with an anti-Hebbian iSTDP rule of Eqs. (7) and (8) is done by employing the Heun method [120] with the time step Δt = 0.01 msec. For each realization of the stochastic process, we choose random initial points [vi (0), ui (0)] for the ith (i = 1, …, N) FS interneuron with uniform probability in the range of vi (0) ∈ (−50, −45) and ui (0) ∈ (10, 15).
III. EFFECT OF INHIBITORY STDP ON FAST SPARSELY SYNCHRONIZED RHYTHMS
We consider the Watts-Strogatz SWN with high clustering and short path length when the rewiring probability p is 0.25. This SWN is composed of N inhibitory Izhikevich FS interneurons. Throughout the paper, N = 103 except for the cases in Fig. 2(a) and in Figs. 4(b1), 4(b2), and 4(c).
A. FSS in The Absence of iSTDP
First, we are concerned about FSS in the absence of iSTDP in the SWN with p = 0.25. The coupling strengths {Jij} are static, and their values are chosen from the Gaussian distribution with the mean J0 (= 700) and the standard deviation σ0 (=5). Population synchronization may be well visualized in the raster plot of neural spikes which is a collection of spike trains of individual interneurons. Such raster plots of spikes are fundamental data in experimental neuroscience. As a collective quantity showing population behaviors, we use an instantaneous population spike rate (IPSR) which may be obtained from the raster plots of spikes [16, 24–29, 121]. For the synchronous case, “stripes” (composed of spikes and indicating population synchronization) are found to be formed in the raster plot, while in the desynchronized case spikes are completely scattered. Hence, for a synchronous case, an oscillating IPSR R (t) appears, while for a desynchronized case R (t) is nearly stationary. To obtain a smooth IPSR, we employ the kernel density estimation (kernel smoother) [122]. Each spike in the raster plot is convoluted (or blurred) with a kernel function Kh (t) to obtain a smooth estimate of IPSR R (t): where is the sth spiking time of the ith interneuron, ni is the total number of spikes for the ith neuron, and we use a Gaussian kernel function of band width h:
Throughout the paper, the band width h of Kh (t) is 1 msec.
The mean square deviation of R(t), plays the role of an order parameter [121]; the over-bar represents the time average. This order parameter may be regarded as a thermodynamic measure because it concerns just the macroscopic IPSR kernel estimate R(t) without any consideration between R (t) and microscopic individual spikes. In the thermodynamic limit of N → ∞, the order parameter approaches a non-zero (zero) limit value for the synchronized (desynchronized) state. Figure 2(a) shows plots of log10〈〉r versus D. In each realization, we discard the first time steps of a stochastic trajectory as transients for 103 msec, and then we numerically compute by following the stochastic trajectory for 3 × 104 msec. Throughout the paper, 〈· · ·〉r denotes an average over 20 realizations. With increasing N up to 104, these numerical calculations for 〈〉r are done for various values of D. For D < D* (≃ 558), synchronized states exist because the order parameter 〈〉r tends to converge toward non-zero limit values. On the other hand, for D > D*, with increasing N the order parameter 〈〉r tends to approach zero, and hence a transition to desynchronization occurs due to a destructive role of noise spoiling the population synchronization.
Figures 2(b1)-2(b6) show raster plots of spikes for various values of D, and their corresponding IPSR kernel estimates R (t) are also shown in Figs. 2(c1)-2(c6). For D = 50, clear (straight) stripes appear successively in the raster plot of spikes, as shown in Fig. 2(b1), and the corresponding IPSR R (t) exhibits a regular oscillation [see Fig. 2(c1)]. However, as D is increased, the raster plot of spikes begins to show a zigzag pattern intermingled with inclined partial stripes of spikes due to local clustering, as shown in Fig. 2(b2) for D = 150, and hence the amplitudes of the IPSR R (t) are reduced so much in comparison with those for D = 50 [see Fig. 2(c2)]. With further increase in D, zigzag stripes in the raster plot are smeared (see the cases of D = 250, 350, and 450), and hence the amplitudes of R (t) decrease. Eventually, when passing D* desynchronization occurs due to overlap of smeared zigzag stripes, as shown in Fig. 2(b6) for D = 750, and then the IPSR R (t) becomes nearly stationary (i.e., no population rhythm appears) [see Fig. 2(c6)].
In the synchronized region for D < D*, we also compare population oscillating behaviors of R (t) with spiking behaviors of individual interneurons. Figure 2(d) shows plots of the population frequency 〈fp〉r (represented by open circles) and the population-averaged MFR 〈〈fi〉〉r (denoted by crosses) of individual interneurons versus D. In each realization, we get fp from the one-sided power spectrum of (the overbar represents the time average) with mean-squared amplitude normalization which is obtained from 216 data points, and also obtain the MFR fi for each interneuron through averaging for 2 × 104 msec; 〈· · ·〉 denotes a population average over all interneurons. As examples, power spectra are shown for various values of D in Figs. 2(e1)-2(e5). Moreover, interspike interval (ISI) histograms for individual interneurons are also given in Figs. 2(f1)-2(f6). For each D, we obtain the ISI histogram via collecting 105 ISIs from all interneurons; the bin size for the histogram is 0.5 msec. For D = 50, R (t) shows a regular oscillation with population frequency 〈fp〉r ≃ 63.8 Hz [see Fig. 2(e1)]. The ISI histogram in Fig. 2(f1) has a single peak, and 〈ISI〉r (average ISI) ≃ 15.7 msec. Hence, individual interneurons fire regularly like clocks with the population-averaged MFR 〈〈fi〉〉r which is the same as 〈fp〉r. For this case, all interneurons make firings in each spiking stripe in the raster plot (i.e., each stripe is fully occupied by spikes of all interneurons). Consequently, full synchronization with 〈fp〉r = 〈〈fi〉〉r occurs for D = 50. This kind of full synchronization persists until D = Dth ≃ 65, as shown in Fig. 2(d). However, when passing the threshold Dth, full synchronization is developed into sparse synchronization with 〈fp〉r > 〈〈fi〉〉r via a pitchfork bifurcation [see Fig. 2(d)].
For the case of sparse synchronization, individual interneurons fire at lower rates than the population frequency, and hence only a smaller fraction of interneurons fire in each spiking stripe in the raster plot (i.e., each stripe is sparsely occupied by spikes of a smaller fraction of interneurons). As D is increased, the occupation degree of spikes (representing the density of spiking stripes in the raster plot) decreases, along with decrease in the pacing degree of spikes (denoting smearing of spiking stripes), which results in decrease in amplitudes of R(t) [see Figs. 2(c2)-2(c5)]. We note that, with increasing D from Dth, the interval between stripes in the raster plot becomes smaller. Hence, the population frequency 〈fp〉r increases monotonically, as shown in Fig. 2(d) [see also Figs. 2(e2)-2(e5)]. As a result, fast sparsely synchronized rhythms appear in the range of Dth < D < D*. On the other hand, the population-averaged MFR 〈〈fi〉〉r is much less than 〈fp〉r due to sparse occupation of spikes in stripes [see Fig. 2(d)]. As a result, spiking behaviors of individual interneurons differs markedly from the fast population oscillatory behaviors.
Hereafter, we pay our attention to FSS (occurring for Dth < D < D*). Unlike the case of full synchronization, individual interneurons exhibit intermittent spikings phase-locked to the IPSR R (t) at random multiples of the global period TG of R (t). This “stochastic phase locking,” leading to “stochastic spike skipping,” is well shown in the ISI histogram with multiple peaks appearing at multiples of TG, as shown in Figs. 2(f2)-2(f5), in contrast to the case of full synchronization with a single-peaked ISI histogram. Similar skipping phenomena of spikings (characterized with multi-peaked ISI histograms) have also been found in networks of coupled inhibitory neurons in the presence of noise where noise-induced hopping from one cluster to another one occurs [71], in single noisy neuron models exhibiting stochastic resonance due to a weak periodic external force [123, 124], and in inhibitory networks of coupled subthreshold neurons showing stochastic spiking coherence [125–127]. Due to this stochastic spike skipping, sparse occupation occurs in spiking stripes in the raster plot. For this case, the ensemble-averaged MFR 〈〈fi〉〉r of individual interneurons becomes less than the population frequency 〈fp〉r, which results in occurrence of sparse synchronization. We also note that, with increasing D, multiple peaks in the ISI histogram begin to merge [see Figs. 2(f2)-2(f5)], and hence spiking stripes in the raster plot become more and more smeared (i.e., pacing degree of spikes in the raster plot decreases). Eventually, in the case of desynchronized states for D > D*, multiple peaks overlap completely [e.g., see Fig. 2(f6) for D = 750], and hence spikes in the raster plot are completely scattered.
We now measure the degree of FSS in the range of Dth < D < D* by employing the statistical-mechanical spiking measure Ms [121]. For the case of FSS, sparse stripes appear successively in the raster plot of spikes. The spiking measure Mi of the ith stripe is defined by the product of the occupation degree Oi of spikes (representing the density of the ith stripe) and the pacing degree Pi of spikes (denoting the smearing of the ith stripe):
The occupation degree Oi of spikes in the stripe is given by the fraction of spiking neurons: where is the number of spiking neurons in the ith stripe. For the case of sparse synchronization, Oi < 1, in contrast to the case of full synchronization with Oi = 1. The pacing degree Pi of spikes in the ith stripe can be determined in a statistical-mechanical way by taking into account their contributions to the macroscopic IPSR R (t). Central maxima of R (t) between neighboring left and right minima of R (t) coincide with centers of stripes in the raster plot. A global cycle begins from a left minimum of R (t), passes a maximum, and ends at a right minimum. An instantaneous global phase Φ(t) of R(t) was introduced via linear interpolation in the region forming a global cycle (for details, refer to Eqs. (16) and (17) in [121]). Then, the contribution of the kth microscopic spike in the ith stripe occurring at the time to R(t) is given by cos Φk, where Φk is the global phase at the kth spiking time [i.e., ]. A microscopic spike makes the most constructive (in-phase) contribution to R (t) when the corresponding global phase Φk is 2πn (n = 0, 1, 2, …). On the other hand, it makes the most destructive (anti-phase) contribution to R (t) when Φk is 2π (n − 1/2). By averaging the contributions of all microscopic spikes in the ith stripe to R (t), we get the pacing degree of spikes in the ith stripe (see Eq. (18) in [121]): where Si is the total number of microscopic spikes in the ith stripe. Then, through averaging Mi of Eq. (12) over a sufficiently large number Ns of stripes, we get the realistic statistical-mechanical spiking measure Ms, based on the IPSR R (t) (see Eq. (19) in [121]):
In each realization, we obtain Ms by following 3 × 103 stripes.
Figures 3(a1)-3(a3) show the average occupation degree 〈〈Oi〉〉r, the average pacing degree 〈〈Pi〉〉r, and the statistical-mechanical spiking measure 〈Ms〉r, respectively. With increasing D from Dth, at first 〈〈Oi〉〉r (denoting the density of stripes in the raster plot) decreases rapidly due to stochastic spike skipping, like the behavior of 〈〈fi〉〉r in Fig. 2(d), and then it tends to approach a limit value (≃ 0.26). The average pacing degree 〈〈Pi〉〉r represents well the smearing degree of stripes in the raster plot [shown in Figs. 2(b2)-2(b5)]. With increasing D, 〈〈Pi〉〉r decreases, and for large D near D* it converges to zero rapidly due to complete overlap of sparse spiking stripes. Through product of the occupation and the pacing degrees of spikes, the statistical-mechanical spiking measure 〈Ms〉r is obtained. Due to the rapid decrease in 〈〈Oi〉〉r, at first 〈Ms〉r also decreases rapidly, and then it makes a slow convergence to zero for D = D*.
So far, we considered the case of p = 0.25. From now on, we fix the value of D at D = 350, and investigate the effect of small-world connectivity on FSS by varying the rewiring probability p. The topological properties of the small-world connectivity has been well characterized in terms of the clustering coefficient and the average path length [95]. The clustering coefficient C, representing the cliquishness of a typical neighborhood in the network, characterizes the local efficiency of information transfer. On the other hand, the average path length L, denoting the typical separation between two nodes in the network, characterizes the global efficiency of information transfer. The Watts-Strogatz SWN interpolates between a regular lattice (corresponding to the case of p = 0) and a random graph (corresponding to the case of p = 1) through random uniform rewiring with the probability p [95]. The regular lattice for p = 0 is highly clustered but large world where L grows linearly with N; C ≃ 0.71 and L ≃ 17.5 for N = 103 [95]. On the other hand, the random graph for p = 1 is poorly clustered but small world where L grows logarithmically with N; C ≃ 0.02 and L ≃ 2.64 for N = 103 [95]. As soon as p increases from zero, L decreases dramatically, which results in occurrence of a small-world phenomenon which is popularized by the phrase of the “six degrees of separation” [98, 99]. However, during this dramatic drop in L, C decreases only a little. Consequently, for small p SWNs with short path length and high clustering appear (e.g., for p = 0.25, C ≃ 0.33 and L ≃ 2.83) [95].
We first consider the population state in the regular lattice for p = 0. As shown in Fig. 4(a1) for N = 103, the raster plot shows a zigzag pattern intermingled with inclined partial stripes of spikes, and R (t) is composed of coherent parts with regular large-amplitude oscillations and incoherent parts with irregular small-amplitude fluctuations [see Fig. 4(a2)]. For p = 0, the clustering coefficient is high, and hence inclined partial stripes (indicating local clustering of spikes) seem to appear in the raster plot of spikes. As N is increased to 104, partial stripes become more inclined from the vertical [see Fig. 4(b1)], and hence spikes become more difficult to keep pace with each other. Consequently, R (t) shows noisy fluctuations with smaller amplitudes, as shown in Fig. 4(b2). Hence, the population state for p = 0 seems to be desynchronized because R (t) tends to be nearly stationary as N increases to the infinity. With increasing p from 0, long-range short-cuts begin to appear, and hence the average path length L becomes shorter. Hence, when global communication between distant neurons are sufficiently efficient, fast sparsely synchronized states may emerge. To examine appearance of FSS, we obtain the order parameter of Eq. (11) by varying p. Figure 4(c) shows plots of the order parameter 〈〉r versus p. For p < p* (≃ 0.12), desynchronized states exist because 〈〉r tends to zero as N is increased. However, when passing the critical value p*, a transition to FSS occurs because the values of 〈〉r become saturated to non-zero limit values.
We now study population and individual behaviors of FSS for p > p*. FSS can be understood well via comparison of population behaviors with individual behaviors. Figures 4(d1)-4(d5) and Figures 4(e1)-4(e5) show raster plots of spikes and IPSR kernel estimates R (t) for various values of p, respectively. As p is increased, the zigzagness degree of partial stripes in the raster plots becomes reduced due to decrease in the clustering coefficient [see Figures 4(d1)-4(d5)]. For (∼ 0.7), the raster plot becomes composed of vertical stripes without zigzag, and then the pacing degree between spikes for becomes nearly the same. As a result, the amplitudes of R (t) increase until , and then they become nearly saturated. For all these values of p, R (t) exhibit regular oscillations with the same population frequency fp ≃ 123 Hz, corresponding to an ultrafast rhythm (100 − 200 Hz). In contrast to this ultrafast rhythm, individual interneurons show intermittent and stochastic discharges like Geiger counters. We collect 105 ISIs from all interneurons, and obtain the ISI histograms which are shown in Figs. 4(f1)-4(f5). Individual interneurons show stochastic phase lockings [i.e., intermittent spikings phase-locked to R (t) at random multiples of the global period TG of R(t)], leading to stochastic spike skipping. Hence, multiple peaks appear at multiples of TG (≃ 8.1 msec) in the ISI histograms. With increasing p, merged multiple peaks begin to be separated, and for nearly the same histograms with clearly separated multiple peaks emerge. Hence, with increasing p the pacing degree between spikes becomes higher, and for it becomes nearly the same. For all these values of p, the population-averaged MFR 〈fi〉, corresponding to the inverse of the average ISI, is 34 Hz. Hence, each interneuron exhibits an average firing sparsely once during about 3.6 population cycles. As a result, for p > p* fast sparsely synchronized rhythms appear.
For p > p*, we characterize FSS by varying p in terms of the average occupation degree 〈〈Oi〉〉r, the average pacing degree 〈〈Pi〉〉r, and the statistical-mechanical spiking measure 〈Ms〉r. Figures 4(g1)-4(g3) show plots of 〈〈Oi〉〉r, 〈〈Pi〉〉r, and 〈Ms〉r, respectively. The average occupation degree 〈〈Oi〉〉r is nearly the same (〈〈Oi〉〉r ≃ 0.28), independently of p; only a fraction (about 1/3.6) of total interneurons fire in each stripe. This sparse occupation results from stochastic spike skipping of individual interneurons which is well seen in the multi-peaked ISI histograms. Hence, 〈〈Oi〉〉r characterizes the sparseness degree of FSS well. In contrast, with increase in p, at first the average pacing degree 〈〈Pi〉〉r makes a rapid increase due to appearance of long-range connections. However, the value of 〈〈Pi〉〉r becomes saturated at because the number of long-range connections which appear up to is enough to obtain maximal pacing degree. As in the case of 〈〈Pi〉〉r, the statistical-mechanical spiking measure 〈Ms〉r increases rapidly up to because 〈〈Oi〉〉r is nearly independent of p. 〈Ms〉r is nearly equal to 〈〈Pi〉〉r /3.6 because of nearly constant sparse occupation [〈〈Oi〉〉r ≃ 1/ 3.6].
B. Effect of iSTDP on FSS
In this subsection, we study the effect of iSTDP on FSS [occurring for Dth (≃ 65) < D < D* (≃ 558) in the absence of iSTDP]. The initial values of synaptic strengths {Jij} are chosen from the Gaussian distribution with the mean J0 (= 700) and the standard deviation σ0 (=5). Then, Jij for each synapse is updated according to a nearest-spike pair-based STDP rule of Eq. (7).
Figure 5(a) shows an asymmetric anti-Hebbian time window for the synaptic modification ΔJij (Δtij) of Eq. (8) versus Δtij). ΔJij (tij) varies depending on the relative time difference between the nearest spike times of the post-synaptic neuron i and the pre-synaptic neuron j. In contrast to the case of a Hebbian time window for the eSTDP [68], when a post-synaptic spike follows a pre-synaptic spike (i.e., Δtij is positive), LTD of synaptic strength appears; otherwise (i.e., Δtij is negative), LTP occurs. A schematic diagram for the nearest-spike pair-based STDP rule is given in Fig. 5(b), where i = 1 and 2 correspond to the post-and the pre-synaptic interneurons. Here, gray boxes denote stripes in the raster plot, and spikes in the stripes are denoted by solid circles. When the post-synaptic neuron (i = 1) fires a spike, LTD (represented by solid lines) occurs through iSTDP between the post-synaptic spike and the previous nearest pre-synaptic spike. On the other hand, when the pre-synaptic neuron (i = 2) fires a spike, LTP (denoted by dashed lines) occurs via iSTDP between the pre-synaptic spike and the previous nearest post-synaptic spike. For the case of sparse synchronization, individual interneurons make stochastic spike skipping (i.e., they make intermittent and stochastic discharges). As a result of stochastic spike skipping, nearest-neighboring pre- and post-synaptic spikes may appear in any two separate stripes (e.g., nearest-neighboring, next-nearest-neighboring or farther-separated stripes), as well as in the same stripe, in contrast to the case of full synchronization where they appear in the same or just in the nearest-neighboring stripes [compare Fig. 5(b) with Fig. 4(b) (corresponding to the case of full synchronization) in [68]]. For simplicity, only the cases, corresponding to the same, the nearest-neighboring, and the next-nearest-neighboring stripes, are shown in Fig. 5(b).
Figure 6(a) shows time-evolutions of population-averaged synaptic strengths 〈Jij〉 for various values of D in the SWN with p = 0.25; 〈· · ·〉 represents an average over all synapses. For each case of D = 150, 250, and 350, 〈Jij〉 decreases monotonically below its initial value J0 (=700), and it approaches a saturated limit value nearly at t = 1000 sec. Consequently, LTD occurs for these values of D. On the other hand, for D = 450 increases monotonically above J0, and approaches a saturated limit value . As a result, LTP occurs for the case of D = 450. Histograms for fraction of synapses versus (saturated limit values of Jij at t = 1000 sec) are shown in black color for various values of D in Figs. 6(b1)-6(b4); the bin size for each histogram is 10. For comparison, initial distributions of synaptic strengths {Jij} (i.e., Gaussian distributions whose mean J0 and standard deviation σ0 are 700 and 5, respectively) are also shown in gray color. For the cases of LTD (D = 150, 250, and 350), their black histograms lie on the left side of the initial gray histograms, and hence their population-averaged values become smaller than the initial value J0. On the other hand, the black histogram for the case of LTP (D = 450) is shifted to the right side of the initial gray histogram, and hence its population-averaged value becomes larger than J0. For both cases of LTD and LTP, their black histograms are much wider than the initial gray histograms [i.e., the standard deviations σJ are very larger than the initial one σ0]. Figure 6(c1) shows a plot of population-averaged limit values of synaptic strengths versus D. Here, the horizontal dotted line represents the initial average value of coupling strengths J0, and the threshold value (≃ 423) for LTD/LTP (where = J0) is represented by a solid circle. Hence, LTD occurs in a larger range of FSS ; FSS in the absence of iSTDP appears in the range of Dth < D < D* (≃ 558). As D is decreased from , decreases monotonically. In contrast, LTP takes place in a smaller range of FSS (i.e., ), and with increasing D from increases monotonically. Figure 6(c2) also shows plots of standard deviations 〈σJ〉r versus D. All the values of 〈σJ〉r are much larger than the initial values σ0 (=5). The effects of LTD and LTP on FSS after the saturation time (t = 1000 sec) may be well shown in the raster plot of spikes and the corresponding IPSR kernel estimate R (t). Figures 6(d1)-6(d4) and Figures 6(e1)-6(e4) show raster plots of spikes and the IPSR kernel estimates R (t) for various values of D, respectively. When compared with Figs. 2(b2)-2(b5) and Figs. 2(c2)-2(c5) in the absence of STDP, the degrees of FSS for the case of LTD (D = 150, 250, and 350) are increased (i.e., the amplitudes of R (t) are increased) due to decreased mean synaptic inhibition. On the other hand, in the case of LTP (D = 450) the population state becomes desynchronized (i.e., R (t) becomes nearly stationary) because of increased mean synaptic inhibition. Due to inhibition, the roles of LTD and LTP in inhibitory synaptic plasticity are reversed in comparison with those in excitatory synaptic plasticity where the degree of population synchronization is increased (decreased) via LTP (LTD) [68].
In the presence of iSTDP, we also characterize individual and population behaviors for FSS (where individual firing activities differ markedly from population oscillatory behaviors) after the saturation time (t = 1000 sec) in the range of Dth (≃ 65) < D < D* (≃ 426) (where FSS persists in the presence of iSTDP). For comparison, corresponding quantities for FSS in the absence of iSTDP are also given in the range of Dth < D < D* (≃ 558) (where FSS appears in the absence of iSTDP). Figure 7(a1) shows plots of the population frequency 〈fp〉r of the IPSR R (t) (open circles) and the population-averaged MFR 〈〈fi〉〉r of individual interneurons (open triangles) versus D; 〈fp〉r (solid circles) and 〈〈fi〉〉r (solid triangles) in the absence of iSTDP are also shown. With decreasing D from D* to Dth, 〈fp〉r and 〈〈fi〉〉r in the absence of iSTDP approach each other (through decrease in 〈fp〉r and increase in 〈〈fi〉〉r), and eventually they merge for D = Dth. As a result, for D ≤ Dth full synchronization with 〈fp〉r = 〈〈fi〉〉r appears in the absence of iSTDP. In the presence of iSTDP, the values of 〈fp〉r (open circles) are larger than those (solid circles) in the absence of iSTDP mainly due to decreased mean synaptic inhibition 〈Jij〉 (i.e., LTD). As D is decreased from D**, 〈fp〉r (open circles) decreases. However, from D ∼ 100 it begins to increase slowly, and then its difference from the value (solid circles) in the absence of iSTDP increases. For D > 150, values of 〈〈fi〉〉r (open triangles) are much larger than those (solid triangles) in the absence of iSTDP mainly because of LTD (i.e., decreased mean synaptic inhibition 〈Jij〉). With decreasing from D = 150, 〈〈fi〉〉r (open triangle) begins to decrease slowly, in contrast to increase in 〈〈fi〉〉r (solid triangle) for the case without iSTDP. For D ∼ 75 they cross, and then the values of 〈〈fi〉〉r (open triangles) become a little smaller than those (solid triangles) in the absence of iSTDP. Consequently, for D = Dth the difference between 〈fp〉r (open circle) and 〈〈fi〉〉r (open triangle) is non-zero (i.e., 〈fp〉r > 〈〈fi〉〉r), in contrast to the case without iSTDP (where the difference becomes zero). This tendency persists for D < Dth, and hence full synchronization (with 〈fp〉r = 〈〈fi〉〉r for 0 ≤ D ≤ Dth) in the case without iSTDP breaks up into FSS (with 〈fp〉r > 〈〈fi〉〉r) in the presence of iSTDP. Figure 7(a2) also shows a plot of the standard deviation 〈σf〉r (open circles) for the distribution of MFRs of individual interneurons versus D; crosses represent 〈σf〉r in the absence of iSTDP. Values of 〈σf〉r (open circles) are larger than those (crosses) in the absence of iSTDP mainly due to increased standard deviation 〈σJ〉r of synaptic strengths. Particularly, near Dth their differences become much larger because 〈σf〉r in the absence of iSTDP tends to converge to zero (i.e., 〈σf〉r = 0 in the case of full synchronization for D = Dth without iSTDP). This big difference in 〈σf〉r near Dth in the presence of iSTDP results from the break-up of full synchronization for D ≤ Dth due to the dominant effect of large dispersions in synaptic inhibition.
For the case of FSS, stochastic phase locking, leading to stochastic spike skipping, is well shown in the ISI histogram with multiple peaks appearing at multiples of the global period TG of the IPSR R (t), as shown in Figs. 7(b1)-7(b3). Due to the stochastic spike skipping, sparse occupation occurs in stripes in the raster plot of spikes. As a result, the ensemble-averaged MFR 〈fi〉 of individual interneurons becomes less than the population frequency fp. In comparison with those for the case without iSTDP [see Figs. 2(f2)-2(f4)], the peaks are shifted a little to the left, and the heights of the 1st and the 2nd peaks are increased. Hence, the average ISI 〈ISI〉 becomes shorter in the presence of iSTDP, which results in increase in the population-averaged MFRs 〈fi〉. For the cases of D = 350 and 250, peaks in the presence of iSTDP are clearer than those in the absence of iSTDP, mainly due to decreased synaptic inhibition 〈Jij〉 (i.e., LTD), and hence the pacing between spikes in the raster plots are increased for the case with iSTDP. On the other hand, for a smaller case of D = 150 a little merging between peaks occurs in the presence of iSTDP, mainly because of the dominant effect of increased standard deviation σJ (overcoming the effect of LTD), and hence the pacing between spikes is decreased a little. For the case of desynchronized states for D > D**, complete overlap between multiple peaks occurs [e.g., see the case of D = 450 in Fig. 7(b4)], and hence spikes in the raster plot are completely scattered, as shown in Fig. 6(d4).
Figures 7(c1)-7(c2) show the average occupation degree 〈〈Oi〉〉r and the average pacing degree 〈〈Pi〉〉r (represented by open circles), respectively; for comparison, 〈〈Oi〉〉r and 〈〈Pi〉〉r (denoted by crosses) are also shown in the case without iSTDP. In most cases of LTD, the values of 〈〈Oi〉〉r (open circles) are larger than those (crosses) in the absence of iSTDP, due to decreased mean synaptic inhibition. With decreasing D from D**, there are no particular variations in 〈〈Oi〉〉r (open circles). On the other hand, 〈〈Oi〉〉r (crosses) in the absence of iSTDP increases rapidly to 1 near Dth, because of existence of full synchronization for D ≤ Dth with 〈〈Oi〉〉r = 1. Hence, near Dth, the values (open circles) of 〈〈Oi〉〉r in the presence of iSTDP are smaller than those (crosses) for the case without iSTDP due to the dominant effect of standard deviations of synaptic inhibition strengths (causing break up of full synchronization in the absence of iSTDP). Next, we consider the average pacing degree 〈〈Pi〉〉r. As D is increased from Dth to D*, 〈〈Pi〉〉r (crosses) in the absence of iSTDP decreases smoothly. On the other hand, 〈〈Pi〉〉r in the presence of iSTDP shows a step-like transition. In the region of LTD, there are no particular variations in 〈〈Pi〉〉r (just a little decrease with increasing D). Near (≃ 423), a rapid transition to the case of 〈〈Pi〉〉r = 0 occurs due to LTP (i.e., increased mean synaptic inhibition), in contrast to the smooth decrease in 〈〈Pi〉〉r (crosses) in the absence of iSTDP. For , the values of 〈〈Pi〉〉r (open circles) are larger than those (crosses) in the case without iSTDP mainly because of LTD (i.e., decreased mean synaptic inhibition). However, for Dth < D < ∼ 200 the values (open circles) of 〈〈Pi〉〉r in the presence of iSTDP are smaller than those (crosses) for the case without iSTDP mainly because of the dominant effect of standard deviations of synaptic inhibition strengths.
The statistical-mechanical spiking measure 〈Ms〉r (combining the effect of both the average occupation and pacing degrees) is represented by open circles in Fig. 7(c3). With decreasing from D* 〈Ms〉r in the absence of iSTDP (denoted by crosses) increases smoothly, and near D ∼100 it begins to increase rapidly due to existence of full synchronization for D = Dth. In contrast, in the presence of iSTDP, 〈Ms〉r shows a step-like transition (see open circles). Due to the effect of 〈〈Pi〉〉r, a rapid transition to the case of 〈Ms〉r = 0 occurs near because of LTP (decreasing the degree of FSS). On the other hand, in most cases of LTD (), the values of 〈Ms〉r (open circles) are larger than those (crosses) in the case without iSTDP mainly because of LTD (increasing the degree of FSS). However, for Dth < D < ∼ 100 the values (open circles) of 〈Ms〉r in the presence of iSTDP are smaller than those (crosses) for the case without iSTDP mainly due to the dominant effect of standard deviations of synaptic inhibition strengths (decreasing the degree of FSS). As a result, in most cases of LTD (increasing the degree of FSS), good synchronization with higher 〈Ms〉r gets better; in some other cases near Dth the degree of good synchronization decreases mainly due to the dominant effect of standard deviation (decreasing the degree of FSS) of synaptic inhibition strengths. On the other hand, in all cases bad synchronization with lower 〈Ms〉r gets worse via LTP (decreasing the degree of FSS). This kind of Matthew effect (valid in most cases of LTD) in inhibitory synaptic plasticity is in contrast to the Matthew effect in excitatory synaptic plasticity where good (bad) synchronization gets better (worse) via LTP (LTD) [68, 69]; the roles of LTD and LTP in the presence of iSTDP are reversed in comparison to those in the case of eSTDP.
From now on, we make an intensive investigation on emergences of LTD and LTP of synaptic strengths via a microscopic method based on the distributions of time delays {Δtij} between the pre- and the post-synaptic spike times. Figures 8(a1)-8(a5) and 8(b1)–8(b5) show time-evolutions of normalized histograms H (Δtij) for the distributions of time delays {Δtij} for D = 350 and 450, respectively; the bin size in each histogram is 0.5 msec. Here, we consider 5 stages, represented by I (starting from 0 sec), II (starting from 100 sec), III (starting from 300 sec), IV (starting from 500 sec), and V (starting from 800 sec). At each stage, we get the distribution of {Δtij} for all synaptic pairs during 0.2 sec and obtain the normalized histogram by dividing the distribution with the total number of synapses (=50000). For D = 350 (LTD), multi-peaks appear in each histogram, in contrast to the case of full synchronization [68]. As explained in Fig. 5(b), due to stochastic spike skipping, nearest-neighboring pre- and post-synaptic spikes appear in any two separate stripes (e.g., nearest-neighboring, next-nearest-neighboring or farther-separated stripes), as well as in the same stripe, which is similar to the multi-peaked ISI histogram. In the stage I, in addition to the main central (1st-order) peak, higher kth-order (k = 2, …, 7) left and right minor peaks also are well seen. Here, LTD and LTP occur in the black (t > 0) and the gray (Δt < 0) parts, respectively. As the time t is increased (i.e., with increase in the level of stage), peaks become narrowed, and then they become sharper. Particularly, heights of major (k = 1, 2, 3, 4) peaks tend to be increased, while those of minor (k = 5, 6, 7) peaks seem to be decreased. Intervals between peaks also seem to be decreased a little because the population frequency fp of R (t) increases a little with the stage. In the stage I, the effect in the right black part (LTD) is dominant, in comparison with the effect in the left gray part (LTP), and hence the overall net LTD begins to emerge. As the level of stage is increased, the effect of LTD in the black part tends to nearly cancel out the effect of LTP in the gray part at the stage V. For D = 450 (LTP), in the initial stage I, multi-peaks are well seen in the histogram, like the case of D = 350. For this initial stage, the effect in the left gray part (LTP) is dominant, in comparison with the effect in the right black part (LTD), and hence the overall net LTP begins to emerge. However, with increasing the level of stage, peaks become wider and the tendency of merging between the peaks is more and more intensified, in contrast to the case of D = 350. Furthermore, the effect of LTP in the gray part tends to nearly cancel out the effect of LTD in the black part at the stage V.
We consider successive time intervals Ik (tk, tk+1), where tk = 0.2 · (k − 1) sec (k = 1, 2, 3, …). With increasing the time t, in each kth time interval Ik, we obtain the kth normalized histogram Hk (Δtij) (k = 1, 2, 3,…) through the distribution of {Δtij} for all synaptic pairs during 0.2 sec. Then, from Eq. (7), we get the population-averaged synaptic strength 〈Jij〉k recursively: where 〈Jij〉0 = J0 (=700: initial mean value), 〈· · · 〉k means the average over the distribution of time delays {Δtij} for all synaptic pairs in the kth time interval, and the multiplicative synaptic modification (Δtij) is given by the product of the multiplicative factor (J* − Jij) [Jij: synaptic coupling strength at the (k − 1)th stage] and the absolute value of synaptic modification |ΔJij (Δtij)|:
Here, we obtain the population-averaged multiplicative synaptic modification for the kth stage via a population-average approximation where Jij is replaced by its population average 〈Jij〉k−1 at the (k − 1)th stage:
Here, 〈|ΔJij (Δtij)|〉k may be easily obtained from the kth normalized histogram Hk (Δtij):
Using Eqs. (16), (18), and (19), we obtain approximate values of and 〈Jij〉k in a recursive way. Figure 8(c) shows time-evolutions of for D = 350 (black curve) and D = 450 (gray curve). for D = 350 is negative, while for D = 450 is positive. For both cases they converge toward nearly zero at the stage V (starting from 800 sec) because the effects of LTD and LTP in the normalized histograms are nearly cancelled out. The time-evolutions of 〈Jij〉 for D = 350 (solid circles) and D = 450 (open circles) are also shown in Fig. 8(d). We note that the approximately-obtained values for 〈Jij〉 agree well with directly-obtained ones [denoted by the gray solid (dashed) line for D = 350 (450)] in Fig. 6(a). Consequently, LTD (LTP) emerges for D = 350 (450).
Finally, we investigate the effect of network architecture on FSS for D = 350 by varying the rewiring probability p in the presence of iSTDP; in the absence of iSTDP, FSS appears for p > p* (≃ 0.12). Figure 9(a) shows time-evolutions of population-averaged synaptic strengths 〈Jij〉 for various values of p. For each case of p = 0.25, 0.4, 0.7, and 1.0, 〈Jij〉 decreases monotonically below its initial value J0 (=700), and it approaches a saturated limit value nearly at t = 1000 sec. As a result, LTD occurs for these values of p. On the other hand, for p = 0.15 〈Jij〉 increases monotonically above J0, and approaches a saturated limit value . As a result, LTP occurs for the case of p = 0.15. Figure 9(b1) shows a plot of population-averaged limit values of synaptic strengths (: saturated limit values of Jij at t = 1000 sec) versus p. Here, the horizontal dotted line represents the initial average value of coupling strengths J0 (= 700), and the threshold value pth (≃ 0.185) for LTD/LTP (where = J0) is represented by a solid circle. Hence, LTD occurs in a larger range of p* > pth, while LTP takes place in a smaller range of p* < p < pth. Figure 9(b2) also shows a plot of standard deviations 〈σJ〉r versus p. All the values of 〈σJ〉r are much larger than the initial value σ0 (=5). The effects of LTD and LTP on FSS after the saturation time (t = 1000 sec) may be well shown in the raster plot of spikes and the corresponding IPSR kernel estimate R (t) which are shown in Figs. 9(c1)-9(c5) and Figs. 9(d1)-9(d5), respectively. In comparison with Figs. 4(d1)-4(d5) and Figs. 4(e1)-4(e5) in the absence of STDP, the degrees of FSS for the case of LTD (p = 0.25, 0.4, 0.7, and 1.0) are increased (i.e., the amplitudes of R (t) are increased) due to decreased mean synaptic inhibition. On the other hand, for the case of LTP (p = 0.15) the population state becomes desynchronized (i.e., R (t) becomes nearly stationary) because of increased mean synaptic inhibition.
Figures 9(e1) and 9(e2) show the average occupation degree 〈〈Oi〉〉r and the average pacing degree 〈〈Pi〉〉r of FSS (represented by open circles), respectively; for comparison, 〈〈Oi〉〉r and 〈〈Pi〉〉r (denoted by crosses) are also shown in the case without iSTDP. In the presence of iSTDP, 〈〈Oi〉〉r (open circles) shows just a little variation, and their values are larger than those (crosses) in the absence of iSTDP, mainly due to LTD. On the other hand, 〈〈Pi〉〉r in the presence of iSTDP shows a step-like transition. In most region of LTD, there are no particular variations in 〈〈Pi〉〉r (just a little decrease with decreasing p), and their values are larger than those (crosses) in the absence of iSTDP mainly because of decreased mean synaptic inhibition. However, near pth (≃ 0.185), a rapid transition to the case of 〈〈Pi〉〉r = 0 occurs due to LTP (i.e., increased mean synaptic inhibition), in contrast to the smooth decrease in 〈〈Pi〉〉r (crosses) in the absence of iSTDP. Figure 9(e3) shows the statistical-mechanical spiking measure 〈Ms〉r (combining the effect of both the average occupation and pacing degrees and represented by open circles) in the range of p** (≃ 0.161) < p ≤ 1 (where FSS persists in the presence of iSTDP). In the absence of iSTDP, with decreasing from p = 1 to p* (≃ 0.12), 〈Ms〉r (denoted by crosses) decreases smoothly. In contrast, in the presence of iSTDP, 〈Ms〉r shows a step-like transition (see open circles). Due to the effect of 〈〈Pi〉〉r, a rapid transition to the case of 〈Ms〉r = 0 occurs near pth because of LTP (decreasing the degree of FSS). On the other hand, in most region of p, the values of 〈Ms〉r (open circles) are larger than those (crosses) in the case without iSTDP, mainly because of LTD (increasing the degree of FSS). As a result, good synchronization with higher 〈Ms〉r gets better via LTD, while bad synchronization with lower 〈Ms〉r gets worse via LTP. This kind of Matthew effect in inhibitory synaptic plasticity is in contrast to the Matthew effect in excitatory synaptic plasticity where good (bad) synchronization gets better (worse) via LTP (LTD) [68, 69].
IV. SUMMARY AND DISCUSSION
We are interested in synchronized brain rhythms in health and disease [1, 2]. For example, synchronous neural oscillations are used for efficient sensory processing such as binding of the integrated whole image in the visual cortex via synchronization of neural firings [128–131] In addition to such neural encoding of sensory stimuli, neural synchronization is also correlated with pathological brain rhythms related to neural disease (e.g., Parkinsons disease, epilepsy, and schizophrenia) [132–135].
Particularly, we are concerned about fast sparsely synchronized rhythms in an inhibitory Watts-Strogatz SWN of Izhikevich FS interneurons. A neural circuit in the major parts of the brain such as thalamus, hippocampus and cortex is composed of a few types of excitatory principal cells and diverse types of inhibitory interneurons. Functional diversity of interneurons increases the computational power of principal cells [1, 70]. When the synaptic decay time is enough long, mutual inhibition may synchronize neural firings. By providing a coherent oscillatory output to the principal cells, the interneuronal networks play the role of the backbones (i.e., pacemakers or synchronizers) for many brain rhythms such as the 10-Hz thalamocortical spindle rhythms [71] and the 40-Hz gamma rhythms in the hippocampus and the cortex [72, 136]. A framework for emergence of sparsely synchronized rhythms (where stochastic and intermittent single-cell firing activity is markedly different from fast population oscillation) was developed in random networks with delayed synaptic connections [24–27]. Each interneuron in the interneuronal network receives stochastic external excitatory synaptic inputs. When this background noise is strong, interneurons discharge irregularly as Geiger counters, and the population state becomes desynchronized. However, as the inhibitory recurrent feedback becomes sufficiently strong, the asynchronous state becomes destabilized, and then a synchronized state with irregular and sparse neural discharges appears. In this way, under the balance between strong external random excitation and strong recurrent inhibition, FSS was found to emerge in the interneuronal network. For this case of FSS, the population frequency fp is ultrafast (i.e. 100 – 200 Hz), while individual interneurons discharge stochastically at much lower rates than fp. This type of fast sparse rhythms was experimentally observed in hippocampal sharp-wave ripples (100 – 200 Hz), associated with memory consolidation, during slow-wave sleep [137, 138] and in cerebellar fast oscillations (∼ 200 Hz), related to fine motor coordination of inhibitory Purkinje cells [139, 140].
In previous works on FSS, synaptic inhibition strengths were static (i.e., inhibitory synaptic plasticity was not considered). On the other hand, in the present work, adaptive dynamics of synaptic inhibition strengths are governed by the iSTDP (which controls the efficacy of diverse computational functions of interneurons). The effects of iSTDP on FSS have been investigated in the SWN with p = 0.25 by varying the noise intensity D. An asymmetric anti-Hebbian time window has been used for the iSTDP update rule, in contrast to the Hebbian time window for the case of eSTDP. Our results on the effects of iSTDP on FSS are well summarized in the diagram in Fig. 10. For the case of FSS, the ISI histogram consists of multiple peaks, due to stochastic spike skipping. These multiple peaks are sharp for the case of good synchronization (with higher spiking measure), while they are merged in the case of bad synchronization (with lower spiking measure) (see the 1st column in Fig. 10). Emergences of LTD and LTP of synaptic inhibition strengths were investigated via a microscopic method based on the distributions of time delays {Δtij} between the nearest spiking times of the pre- and the post-synaptic interneurons. Like the case of multi-peaked ISI histogram, sharp multi-peaks appear in the normalized histogram for the distribution of {Δtij}; the heights of peaks for the case of good synchronization are higher than those in the case of bad synchronization. For the case of good synchronization, the effect of causal part with Δt > 0 is dominant, and hence LTD begins to occur (see the 2nd column in Fig. 10). On the other hand, in the case of bad synchronization the effect of acausal part with Δt < 0 is dominant, and hence LTP begins to take place.
The distribution of synaptic inhibition strengths {Jij} is evolved in the presence of iSTDP (see the 3rd column in Fig. 10). For the case of good synchronization, its mean 〈Jij〉 is decreased (i.e., LTD occurs), while in the case of bad synchronization 〈Jij〉 is increased (i.e., LTP occurs). The standard deviations σJ for both cases of good and bad synchronization increase. Decrease (increase) in the mean 〈Jij〉 [i.e., LTD (LTP)] favors (disfavors) FSS due to increased (decreased) population-averaged MFR 〈fi〉 of individual interneurons. Increased standard deviation σJ leads to increase in variation of inhibitory synaptic inputs to individual interneurons, and hence distributions of MFRs of individual interneurons become broader (i.e., standard deviation σf for the distribution of MFRs increases). Due to increased σf, it becomes difficult for interneurons to keep their pacing, which results in decrease in the degree of FSS (i.e., FSS is spoiled), as in the case of increasing the noise intensity D. In this way, dispersion of synaptic inhibition strengths seems to play a role which is similar to that of noise.
For the case of iSTDP, in addition to the effect of mean value (LTP or LTD), the effect of standard deviation σJ on population synchronization may also become significant in some cases, in contrast to the case of eSTDP where the mean of LTP/LTD was found to be always dominant [68, 69]. For most cases of good synchronization, the effect of LTD (increasing the degree of FSS) is dominant in comparison with the effect of standard deviation σJ (decreasing the degree of FSS). Consequently, in most cases of good synchronization, it has been found to get better via LTD; in some other cases where the effect of standard deviation σJ is dominant (occurring near Dth), the degree of good synchronization decreases even in the presence of LTD. In contrast, for all cases, bad synchronization has been found to get worse via LTP (decreasing the degree of FSS). This type of Matthew effect (valid in most cases of LTD) in inhibitory synaptic plasticity is in contrast to the Matthew effect in excitatory synaptic plasticity [68, 69]. For the case of eSTDP, LTP (LTD) has a tendency to favor (disfavor) synchronization via positive (structural) feedback, while LTD (LTP) for the case of iSTDP has a tendency favoring (disfavoring) FSS through a negative (structural) feedback. Hence, due to inhibition via negative feedback, the roles of LTD and LTP in inhibitory plasticity are reversed in comparison with those in excitatory synaptic plasticity through positive feedback where good (bad) synchronization gets better (worse) via LTP (LTD). Consequently, in most region of LTD, the degree of FSS becomes increased, and a rapid transition from FSS to desynchronization occurs via LTP, in contrast to the relatively smooth transition in the absence of iSTDP.
The process of iSTDP may be well visualized in the normalized histogram of H (Δtij) and the ISI histogram (see the 4th and 5th columns in Fig. 10). With increasing time t, peaks in the normalized histogram H(Δtij) becomes narrowed and sharper for the case of LTD, while in the case of LTP peaks become wider and merged. After a sufficient time, the effect of LTD in the right causal part with Δt > 0 nearly cancels out the effect of LTP in the left acausal part with Δt < 0. Then, saturated limit states appear without further change in synaptic strengths. For most cases of good synchronization, peaks in the ISI histogram become clearer (i.e., narrower and sharper), mainly due to the dominant effect of decreased synaptic inhibition (i.e., LTD), and hence the pacing between spikes in the raster plot is increased. As a result, in most cases good synchronization gets better via LTD. In contrast, for the case of bad synchronization, complete overlap between peaks in the ISI histogram occurs (i.e., a broad single peak appears) due to increased synaptic inhibition (i.e., LTP), and hence spikes are completely scattered in the raster plot. Consequently, bad synchronization gets worse via LTP.
Emergences of LTD and LTP of synaptic inhibition strengths were investigated via a microscopic method based on the distributions of time delays {Δtij} between the nearest spiking times of the pre- and the post-synaptic interneurons. Time evolutions of normalized histograms H (Δtij) were followed for both cases of LTD and LTP. Using a recurrence relation, we recursively obtained population-averaged synaptic inhibition strength 〈Jij〉 at successive stages through an approximate calculation of population-averaged multiplicative synaptic modification of Eq. (18), based on the normalized histogram at each stage. These approximate values of 〈Jij〉 have been found to agree well with directly-calculated ones. Consequently, one can understand clearly how microscopic distributions of {Δtij} contribute to 〈Jij〉 or more directly to .
By varying the rewiring probability p in the SWN, we also studied the effect of network architecture on FSS in the presence of iSTDP. As in the above case of variation in D for p = 0.25, a Matthew effect has also been found to occur in the case of variation in p for D = 350. As a result, good (bad) synchronization with higher (lower) spiking measure Ms for p > (<)pth (≃ 0.185) gets better (worse) via LTD (LTP).
Finally, we discuss limitations of our work and future works. In our work, we employed the standard “duplet” STDP model, based on the nearest pre- and post-synaptic spike pairs. However, unfortunately this pair-based STDP model accounts for neither the dependence of plasticity on the repetition frequency of the pairs of pre- and post-synaptic spikes, nor the results of recent triplet and quadruplet experiments [141, 142]. Hence, as a future work, it would be interesting to study the effect of iSTDP on FSS by using a triplet iSTDP rule and to compare its results with those for the case of duplet iSTDP rule. Inhibitory neurons have been found to possess diverse types of plasticity rules. For example, Hebbian STDP [143, 144], anti-Hebbian STDP [82, 144–146], and anti-Hebbian STDP with only LTD [79] were observed in the case of excitatory (E) to inhibitory (I) connection. Moreover, for the case of I to E connection, anti-symmetric Hebbian STDP [92] and symmetric (non-Hebbian) STDP [93] were found. However, in the preset work, for simplicity we assumed that all interneurons exhibit identical anti-Hebbian STDP for the case of I to I connection. To take into consideration heterogeneity on synaptic plasticity of inhibitory cells seems to be beyond the present work, and it will be left as a future work. In the present work, we considered only the interneuronal network. As explained above, a major neural circuit consists of two excitatory and inhibitory populations. In previous works [26, 27], they also considered the two-population network with four (I to I, I to E, E to I, and E to E) types of connections. The additional I to E, E to I, and E to E connections have tendency to decrease the population frequency fp. It was found that fp was much reduced to about 30 – 100 Hz (corresponding to gamma rhythms), when compared with the case of pure interneuronal network. Hence, in future, it would be interesting to study the effects of interpopulation (I to E and E to I) STDP on FSS in the two-population network of inhibitory Izhikevich FS interneurons and excitatory Izhikevich regular-spiking neurons, in addition to the studied intrapopulation (E to E and I to I) STDP. In our work, we also considered just the spiking neurons. In addition to spiking, bursting is also another type of neuronal firing activities. Burstings occur when neuronal activity alternates, on a slow timescale, between a silent phase and an active (bursting) phase of fast repetitive spikings. There are several representative examples of bursting neurons. Recently, we also investigated the effect of iSTDP on burst synchronization in a scale-free neuronal network of inhibitory Hindmarsh-Rose bursting neurons [147]. Thus, the effect of iSTDP on burst synchronization was also found to be in contrast to the effect of eSTDP on burst synchronization, similar to the case of spiking neurons. Finally, we note that there exist some limitations to STDP in views of biological contexts [148]. For example, STDP has been questioned as a general model of synaptic plasticity [149], the classic STDP windows for LTP and LTD were found to be only one of many possible ones [150], and because of the attenuation of the back-propagating action potential, STDP was found to depend on the dendritic synapse location [151–153]. In the presence of these limitations, we expect that our results on inhibitory synaptic plasticity of I to I connections would make some contributions for understanding the effects of iSTDP on fast sparsely synchronized rhythms
Acknowledgments
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 20162007688).
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