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Edited by: David Golomb, Ben-Gurion University of the Negev, Israel

Reviewed by: Maoz Shamir, Ben-Gurion University of the Negev, Israel; Florence Isabelle Kleberg, RIKEN Brain Science Institute, Japan

*Correspondence: Sachin S. Talathi

†Present Address: Shivakeshavan Ratnadurai-Giridharan, Mathematical Sciences, Indiana University-Purdue University, Indianapolis, IN, USA

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We investigate the emergence of in-phase synchronization in a heterogeneous network of coupled inhibitory interneurons in the presence of spike timing dependent plasticity (STDP). Using a simple network of two mutually coupled interneurons (2-MCI), we first study the effects of STDP on in-phase synchronization. We demonstrate that, with STDP, the 2-MCI network can evolve to either a state of stable 1:1 in-phase synchronization or exhibit multiple regimes of higher order synchronization states. We show that the emergence of synchronization induces a structural asymmetry in the 2-MCI network such that the synapses onto the high frequency firing neurons are potentiated, while those onto the low frequency firing neurons are de-potentiated, resulting in the directed flow of information from low frequency firing neurons to high frequency firing neurons. Finally, we demonstrate that the principal findings from our analysis of the 2-MCI network contribute to the emergence of robust synchronization in the Wang-Buzsaki network (Wang and Buzsáki,

Cortical gamma rhythms (30–80 Hz) are correlated with diverse brain functions such as memory formation (Singer and Gray,

Interneurons play critical roles for gamma rhythm generation in both mechanisms. However, in ING, interneurons are solely responsible for gamma frequency activity (Bartos et al.,

Previous theoretical works have shown that synchronization is possible in interneuronal networks (Wang and Rinzel,

Evidence from literature suggests a strong correlation between Hebbian learning and Gamma rhythms (Miltner et al.,

In our earlier works, we demonstrated that inhibitory plastic synapses could improve in-phase synchronization in the presence of firing rate heterogeneity in a simple network of uni-directionally coupled interneurons (UCI) (Talathi et al.,

The organization of the paper is as follows: the Methods Section presents the mathematical model for the network, the interneuron, and the synapse used in our studies. We analyze two types of heterogeneity in the interneuronal networks: (1) temporal heterogeneity; which corresponds to the heterogeneity in the intrinsic firing rates of the coupled interneurons, and (2) structural heterogeneity; which corresponds to the heterogeneity in the synaptic coupling strength. We introduce a few network measures in order to quantify the network state in terms of its connectivity and/or synaptic strengths. We then introduce a novel method for using STRC's with a strong second order component to estimate the spike times for a periodically firing neuron that receives periodic synaptic inputs. Finally, we introduce the empirical STDP rule (Haas et al.,

In the Results Section, we systematically analyze the synchronization domain of a 2-MCI system in terms of temporal and structural heterogeneity. We show how heterogeneity and initial values of synaptic strengths determine the final synchronization state of the 2-MCI system. The 1:1 domain of synchronization identified by this analysis allows us to determine how STDP must evolve synaptic strengths in order to achieve 1:1 synchronization. We then use the STDP rule and STRC of the interneuron model to derive a nonlinear map for the evolution of the 2-MCI system to a state of in-phase synchronization. Finally, we demonstrate that the principal findings from our analysis of the 2-MCI network hold for larger MCI networks.

Each neuron in the network of all-to-all coupled inhibitory interneurons is modeled based on a single compartment model for parvalbumin positive inhibitory neuron developed by Wang and Buzsáki (

where ^{2}. _{j}(

where, the parameter ^{DC}-values (and correspondingly, firing periods), as given in Equation (2). Hence for a larger _{ref} = 1 μA/cm^{2} in Equation (2) is set such that the mean intrinsic frequency of firing for the neurons in the network is ≈ 60 Hz (Wang and Buzsáki, _{r} (_{I}, is the reversal potential of the fast GABAergic inhibitory synapse. _{r} (_{∞} = α_{m}∕(α_{m} + β_{m}). The inactivation variable for sodium channel _{X} and β_{X} are given by:

The strength of inhibitory synaptic coupling from the pre-synaptic neuron _{ij}(

where |η| ≤ 100. For all the results presented unless otherwise stated, _{0} = 0.1 mS/cm^{2}. The parameter η regulates the structural heterogeneity in the network such that for η ≠ 0, there is asymmetry in the synaptic coupling strengths of the two mutually coupled neurons. By design, the asymmetry is such that for η > 0, the strength of synapse from neuron with lower index is smaller than that of the synapse from neuron with higher index label. The variable _{ij} defines the topology of the inhibitory neuronal network such that, for an all-to-all coupled network, we have _{ij} = 1 ∀_{ij} = 0 otherwise for all _{i}(

The kinetic equation for _{i}(_{0}(_{0}(

In order to quantify the influence of synaptic plasticity on the structure of synaptic connectivity in the network, we define the following network measures:

(a) Link imbalance: The difference in synaptic strengths of the two coupled neurons,

(b) Neuronal strength: The sum of synaptic strength of all outgoing synapses from the neuron

For all the networks considered here, the neurons in the network are labeled in ascending order of input DC current such that label 0 is assigned to the neuron with lowest intrinsic firing rate while label

As a measure of the influence of synaptic input on the firing times of a given neuron _{i0} is the intrinsic period of spiking, and _{ij} represents the length of the jth spiking cycle from the cycle _{i0}. A synaptic perturbation can either advance or delay the occurrence of an impending spike depending on the bifurcation character of the neuron (Ermentrout, _{R}, the synaptic decay time τ_{D}, the reversal potential of the synapse _{R}, and the synaptic conductance _{R} = −75 mV), as considered in this study, a synaptic input delays the time of occurrence of subsequent spike such that Φ_{i1} ≥ 0 ∀δ_{i2} > 0 for δ_{i0}, and Φ_{ij} = 0 ∀_{R}, τ_{D}, _{R}, and the intrinsic firing period of neuron _{0} and define Φ(τ_{R}, τ_{D}, _{R}, _{0},

We begin by estimating STRC's using the direct method as follows: consider neuron _{i0}. The neuron is perturbed through an inhibitory synapse at time δ_{1}, representing the first cycle after perturbation of length _{i1} ≠ _{i0}. Depending on the properties of the synapse, i.e., _{s}, τ_{R}, τ_{D}, and _{R}; the length of subsequent cycles might change. In Figure _{i1} (solid black line) and Φ_{i2} (solid magenta line), estimated by using the procedure described above for δ_{i0}). We note the presence of a non-zero second order STRC component for the choice of the parameters of the neuron.

_{R} = −75 mV, τ_{R} = 0.1 ms, τ_{D} = 5 ms, and ^{2}. The original STRC is shown in solid colors while the STRC with the included phase correction term (δ^{*}) is represented by the dotted lines. _{t} during every firing cycle of neuron 1. The solid blue lines indicate the spike-time shift calculated using the STRC that incorporates the correction factor. The dotted blue lines indicate the spike-time shift calculated using the original STRC's without any correction term.

In the direct method for computing STRC's, we use a single perturbation to observe the changes in spike times of a periodically firing neuron. These changes in spike times can be used to predict future spike times. If Φ_{i2} ≈ 0, then a single perturbation arriving at δ_{0}, can be assumed to produce a spike time shift such that the next spike occurs at time _{1} = _{0} + _{i0}(1 + Φ_{i∞}(_{i∞}(_{i1}(_{i2}(_{i2}) is present, Φ_{i∞}(

We demonstrate this issue using a 2-UCI network as shown in Figure _{i1} = _{i0}(1 + Φ_{i∞}(

Previously, Talathi et al. (

where, _{00} and _{10} represent the firing periods of neurons 0 and 1, respectively, Φ_{1∞}(_{01}, δ_{s}) is the STRC calculated for the steady-state perturbation time δ_{s}, which is effectively the spike time difference between the two neurons. The spike time difference could also be measured directly by observing simulation results of the 2-UCI network in a state of 1:1 synchrony, which we denote as δ^{*}. When the STRC's second order component is weak, the theoretical predictions match closely with the simulations and we get _{s} and simulation's δ^{*}. We term this error as

In order to characterize _{s}, we need to observe the 2-UCI network in 1:1 synchrony across a range for δ^{*} and δ_{s}. Since, δ_{s} and δ^{*} are spike time differences of the theoretical map solution and simulations, respectively for a particular configuration of the 2-UCI network in 1:1 synchrony, they cannot be arbitrarily predetermined. Instead, by varying the temporal heterogeneity parameter ^{*}, from simulations and the theoretically estimated spike time difference, δ_{s}, between the two neurons. By noting δ_{s}, δ^{*}, and the corresponding error _{s} (see Figure _{s} is approximately linear and can be approximated via a linear fit as _{s} in order for the STRC based solution to match simulation results (see Figure _{s} to any perturbation δ

^{*} and STRC based map predicted δ_{s} as _{s} is due to the inaccuracy of the standard STRC method arising from second order STRC components. _{s} as:

We will assume that all further references to STRC's in this paper incorporate the correction term δ^{*} that is calculated from the given δ

We will consider the spike timing dependent plasticity (STDP) rule for GABAergic synapses, identified at GABAergic synapses in the entorhinal cortex (Haas et al.,

where _{+} ≥ _{−}. The parameters are β = 10 and α = 0.94, which provide a temporal window of ±20 ms over which the efficacy of the synaptic plasticity is non-zero (Haas et al., _{+} = _{−} = 0.01 and we assume an additive (linear) update rule for the modification of the synaptic strengths in the network (Caporale and Dan,

_{+} = _{−} = 0.01 mS/cm^{2}.

We will measure the degree of synchronization _{i}(_{s}) where _{s} is the sampling time interval and _{s}, we define:

where _{V} and σ_{Vi}

The properties of _{i}(

In this section, we investigate the structure of the synchronization manifold of the 2-MCI network as a function of the network heterogeneity parameters: {_{0, i} (_{1, j} (_{n} = _{1,n} − _{0,m} between the nearest spike times of the two neurons. The index _{1,n−1} ≤ _{0, m} ≤ _{1,n}. For a given value of _{0}〉 and 〈_{1}〉. If there exists

we label the point {_{n} approached a fixed value δ_{s} ≤ 〈_{X}〉 representing a phase-locked state (see Figures

In Figures _{n} as function of

In summary, the key conclusion from this analysis is that the 2-MCI network can sustain 1:1 synchronization for a wide range of heterogeneity in the intrinsic firing rates of the coupled neurons for an appropriate choice of structural heterogeneity in the network, i.e., by increasing strength of neuron 0 relative to the strength of neuron 1 and decreasing the sensitivity of neuron 0 relative to that of neuron 1, to trigger an effective structural imbalance η < 0.

We will now investigate the effect of STDP on the dynamics of a heterogeneous 2-MCI network. For all the results presented in this section, we will begin with a structurally homogeneous network i.e., initial η = η_{0} = 0 and investigate the influence of STDP on the network dynamics for various values for _{i} (_{01} = _{10} = _{0}∕2 and (2) with STDP learning, following the additive rule given in Equation (11). The initial synaptic conductance values are set to be the same as for the static case. For each simulation run, after the network has evolved into an asymptotic state (asynchronous firing or the state of m:n synchronization) we estimate the ratio

_{0} = 0. (A)

In the presence of STDP, the synaptic coupling strengths evolve such that, the range of heterogeneity values over which the network can sustain stable states of 1:1 synchronization (

_{0} = 0,

In order to understand how the structural properties of the 2-MCI network change as the network evolves to a state of 1:1 in-phase synchronization under the influence of STDP, in Figure _{0} = 0. The STDP learning begins at time

Starting from η_{0} = 0, the network evolves vertically downwards in terms of the two-dimensional plot of the synchronization manifold in Figure _{1}〉 < 〈_{0}〉. Thus, after STDP is active, the synapse _{01} (with neuron 1 being the post-synaptic neuron for the synapse) increases more often than it decreases and consequently η fluctuates with an increased bias toward η < 0 until the network evolves into the domain of 1:1 synchronization (see Figure _{1}〉 = 〈_{0}〉. Since Δ

We can see from Figure

The synchronization manifold of the static 2-MCI network (Figure _{0} = 0 toward a structurally imbalanced state with η < 0 in order to compensate for _{0} = 0 and correspondingly the likelihood of the network to evolve to 1:1 synchronization decreases, as is evidenced from Figure

In this section, we analyze the stability of the emergent 1:1 in-phase synchronization in the 2-MCI network under the influence of STDP by deriving a nonlinear map for the evolution of the time lag δ between successive spike times of the two neurons in the 2-MCI network using the framework of STRC's. We consider the specific case of the network configuration with parameters: _{0} = 0. Following from the results presented in Section 3.1, for the choice of network heterogeneity and imbalance parameters, in the absence of STDP, the two neurons in the 2-MCI network will fire asynchronously, with mean firing rate of neuron 1 greater than that of neuron 0. Invoking STDP in this situation will, on average, cause the strength _{0} of neuron 0 to increase, while at the same time it will cause the sensitivity _{0} of neuron 0 to decrease. This is because, more often than not, the fast firing neuron 1 will emit more than one spike during each period of spiking from neuron 0. Each firing of neuron 1 (the post-synaptic neuron to synapse from neuron 0 to neuron 1), in turn, increases the strength of synapse _{01} through the STDP rule. By the same token, the strength of synapse _{10} will decrease creating an effective network imbalance η < 0. Thus, starting from a structurally balanced network _{net} = 0, the network will evolve vertically downwards in the two dimensional η − _{0} = 0, the network will eventually evolve into the domain of 1:1 synchronization.

In order to understand how the synaptic strengths evolve within the domain of 1:1 synchronization to the final stable state of 1:1 in-phase synchronization, we use the mathematical framework of STRC's to derive a nonlinear map for the evolution of the time-lag δ and the synaptic strengths _{ij} (_{0}〉 = 〈_{1}〉, the phase locked state remains quasi-static as the synaptic strength continues to evolve toward the asymptotic state of 1:1 in-phase synchronization. In Figure _{0,n} and _{1,n} represents the timing of nth spike originating from neuron 0 and neuron 1 in the 2-MCI network, respectively, then the subsequent spike from the two neurons will emerge at times: _{0,n+1} and _{1,n+1} such that:

where _{ij, n} is the conductance of synapse from the pre-synaptic neuron i to the post-synaptic neuron j, immediately after nth spike is emitted from the post-synaptic neuron. We further assume that STDP induced change in synaptic conductance is instantaneous. The nonlinear map for the evolution of the time-lag δ_{n} = _{1,n} − _{0,m} (index m corresponds to the mth spike from neuron 0 such that _{1, n−1} ≤ _{0, m} ≤ _{1, n}) and the synaptic conductances _{ij, n} can be obtained from Equation (14) as:

where

_{01}, _{10}} due to the STDP rule are also shown. We use this concept of 1:1 phase locked spike-time and STDP evolution to derive the map defined in Equation (15).

To determine whether Equation (15) can predict 1:1 in phase synchronization for the 2-MCI network under the influence of STDP learning, we apply the discrete map for the specific cases of the 2-MCI network with heterogeneity _{01, 0} = _{10, 0} = _{0}∕2 corresponding to the case η_{0} = 0 and δ_{0} = 0.

The results are presented in Figure _{s} = 18.8, α_{s} = 0.1 ms, _{01} = 0.073 mS/cm^{2} and _{10} = 0.027 mS/cm^{2}. The mean period of the synchronized network (described in Section 3.1) is 〈_{0}〉 = 〈_{1}〉 = 18.9 ms. For the special case of 1:1 synchrony, we will refer to the network period as:

Upon simplifying (Equation 15) for the 1:1 steady-state, one can derive a straightforward relationship between δ_{s} and 〈_{0∕1}〉 as: 〈_{0∕1}〉 = δ_{s} + α_{s}. We see that when α_{s} ≈ 0 then 〈_{0∕1}〉 = δ_{s} which is the in-phase 1:1 solution. Though the discrete map correctly predicts the existence of a steady state fixed point solution, a minor discrepancy in the discrete map solution for steady state value of η exists. There is a difference in η of ≈4 between the map's solution and to the solution obtained by numerically solving Equation (1) under the influence of STDP. This discrepancy stems from the linear approximation of the correction factor

For the case

We next determine if the 1:1 in phase is stable. First, we performed linear stability analysis of the discrete map in Equation (15) for the fixed point in-phase solutions {δ_{s}, α_{s}, _{01s}, _{10s}} = {18.8, 0.1, 0.073, 0.027}. We find that all the eigenvalues are real and {λ_{δ}, λ_{α}} < 1, indicating stability. However, {λ_{g01}, λ_{g10}} = 1. These eigenvalues suggest that while the in-phase solution is stable for static synapses, with STDP the system may be marginally stable or unstable. In order to determine if the system was marginally stable at the in-phase solutions, we examined the Jordan form of the linearized state matrix and observed that the Jordan blocks corresponding to unit eigenvalues were scalar. This served as theoretical confirmation for local stability of the system with STDP present.

Additionally, we numerically analyzed the sensitivity of the in-phase solutions reached through STDP. We tested the sensitivity of the solutions by varying the initial conditions of the state variables in Equation (15). In Figure _{0∕1}〉 = δ_{S} for different initial values of η_{0} and _{0} = 0 and even η_{0} = 40, 〈_{0∕1}〉 evolves to the stable in-phase period of 18.9 ms, which is the exact in-phase period predicted by the 2-MCI simulations. The map stably evolves to the in-phase solution even for an initial value that is 200% greater than the solution point. We also confirm that 〈_{0∕1}〉 evolves to the in-phase period for different initial spike time lag (δ_{0}) values of {0, 4, 8, 12} ms. This is illustrated in Figure

_{0∕1}〉 for different initial conditions of η_{0}. _{0∕1}〉 for different initial conditions of δ.

In this section we examine how STDP allows larger MCI networks to synchronize. We consider a homogeneous all-to-all coupled network of 100 neurons with temporal heterogeneity

In Figure

We also look at the other network metrics that we defined in the methods section. In Figure _{ij} = _{ij}_{ij} − _{ji}_{ji}, where _{ij}] ∈ [0, 1] refers to the connectivity between neuron _{ij} > 0 and for _{ij} < 0. This indicates that STDP evolves such that the strength of synapses from slower neurons onto faster neurons are larger in value. This observation is further reinforced by the neuronal strength metric (

In this study we investigated if STDP in mutually coupled interneuronal networks can induce stable in-phase synchronization at gamma frequencies. We first investigated the domains of synchronization for a 2-MCI network in terms of structural and temporal heterogeneity, given by {η, _{01} increasing while _{10} decreases. This in turn, indicated that, for 1:1 and in-phase synchrony, the slower neuron 0, suppresses the faster neuron 1, such that their periods match. In the case of higher order synchronization such as 2:1 synchronization, the faster neuron fires more than once for every single time the slower neuron fires. Here, the period of the slower neuron is significantly increased due to suppression by the faster neuron.

Previous work by White et al. (

In other works, STRC's have been used to calculate future spike times based on periodic synaptic perturbations from other neurons. Most of these studies assumed that the STRC's displayed weak second order components. For example in the work done by Talathi et al. (_{i∞}(δ_{i1}(δ_{i2}(δ_{i∞} term would lead to incorrect predictions.

Recently, Talathi et al. (_{t} and error in spike-times _{i∞} to more precisely describe spike time shifts.

In this work, we performed an in-depth analysis on the synchronization of 2-MCI network and eventually, a 100-MCI network. We utilized the same STDP learning rule form as used by Talathi et al. (

We began our study on the 2-MCI network by identifying the synchronization domain in terms of the structural and firing rate heterogeneity parameters {η,

In the presence of STDP, for η_{0} = 0, the 2-MCI network was found to evolve to in-phase synchronization for a significant range of _{0}, STDP could variably evolve the network to different orders of synchrony.

In order to determine if the 1:1 and in-phase synchronization states of the network were stable, we mathematically analyzed the 2-MCI network and derived a STRC based map for predicting 1:1 and in-phase synchronization. We first validated the map by ensuring that the predicted spike time differences ({δ, α}) were the same as that of the static 2-MCI simulations, within the 1:1 synchronization domain. We then observed the map's evolution of η with STDP for an initial η_{0} = 0 and _{s}. Having validated that the map behaves closely to the 2-MCI simulations, we performed linear stability analysis on the discrete mathematical map expressed in Equation (15). Our findings indicated that the eigenvalues for the spike times {δ, α} were stable (< 1). However, the eigenvalues corresponding to the STDP evolution of {_{01}, _{10}} were marginally stable. To prove that system was at least locally stable, we performed further numerical analyses. Specifically, we examined the evolution of the map with STDP for different initial conditions of δ and η. We observed that the map's predicted 1:1 synchronization period (〈_{0∕1}〉 ≈ δ) and confirmed that it evolved to the period of the 2-MCI simulation in-phase synchronization, for a range of initial conditions. In fact η had to be increased significantly (η ≥ 20) beyond the stable value η = −40 in order for the map to fail to evolve to the in-phase synchronization regime.

We next examined how the scale of MCI networks affects STDP induced synchronization. We constructed and simulated a 100-MCI structurally homogeneous network (η_{0} = 0) with and without STDP for

All of our results suggest that STDP is a very viable mechanism for the formation of interneuronal gamma (ING) oscillations. The original firing rates of the free running interneurons (

ST, PK conceived the initial design of the work with additional input from SR. SR performed all the simulation experiments and analysis. SR wrote the manuscript with significant inputs from PK and ST.

This research was funded by startup funds to ST from the Department of Pediatrics at the University of Florida; the intramural grant on Computational Biology at the University of Florida; and the Wilder Center of Excellence for Epilepsy Research and the Children's Miracle Network. PK was partially supported by the Eckis Professor Endowment at the University of Florida.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.