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Robust Rhythmogenesis Via Spike Timing Dependent Plasticity

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Robust Rhythmogenesis Via Spike Timing Dependent Plasticity bioRxiv preprint doi: https://doi.org/10.1101/2020.07.23.217026; this version posted March 31, 2021. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. Robust Rhythmogenesis via Spike Timing Dependent Plasticity Gabi Socolovsky1, 2, ∗ and Maoz Shamir1, 2, 3 1Department of Physics, Faculty of Natural Sciences, 2Zlotowski Center for Neuroscience, 3Department of Physiology and Cell Biology, Faculty of Health Sciences, Ben-Gurion University of the Negev, Be'er-Sheva, Israel (Dated: March 31, 2021) Rhythmic activity has been observed in numerous animal species ranging from insects to humans, and in relation to a wide range of cognitive tasks. Various experimental and theoretical studies have investigated rhythmic activity. The theoretical efforts have mainly been focused on the neuronal dynamics, under the assumption that network connectivity satisfies certain fine-tuning conditions required to generate oscillations. However, it remains unclear how this fine tuning is achieved. Here we investigated the hypothesis that spike timing dependent plasticity (STDP) can provide the underlying mechanism for tuning synaptic connectivity to generate rhythmic activity. We addressed this question in a modeling study. We examined STDP dynamics in the framework of a network of excitatory and inhibitory neuronal populations that has been suggested to underlie the generation of oscillations in the gamma range. Mean field Fokker Planck equations for the synaptic weights dynamics are derived in the limit of slow learning. We drew on this approximation to determine which types of STDP rules drive the system to exhibit rhythmic activity, and demonstrate how the parameters that characterize the plasticity rule govern the rhythmic activity. Finally, we propose a novel mechanism that can ensure the robustness of self-developing processes, in general and for rhythmogenesis in particular. I. INTRODUCTION Since the cross-correlation of neural activity is central to STDP dynamics, we next define the network dynam- Rhythmic activity in the brain has been observed for ics and analyze its phase diagram and the dependence more than a century [1, 2]. Oscillations in different fre- of the correlations on the synaptic weights. Using the quency bands have been associated with different cogni- separation of timescales in the limit of slow learning we tive tasks and mental states [2{6]. Specifically, rhyth- analyze STDP dynamics and investigate under what con- mic activity in the Gamma band has been described in ditions STDP can stabilize a specific rhythmic activity association with sensory stimulation [7], attentional se- and how the characteristics of the STDP rule govern the lection [8, 9], working memory [10] and other measures resultant rhythmic activity. Finally, we summarize our [11]. Deviation from normal rhythmic activity has been results, discuss possible extensions and limitations and associated with pathology [12{15]. propose a general principle for robust rhythmogenesis. Considerable theoretical efforts have been devoted to unraveling the neural mechanism responsible for gener- ating rhythmic activity in general [2] and in the Gamma II. STDP DYNAMICS band in particular [16{21]. One possible mechanism is based on delayed inhibitory feedback [21{24]. The basic The basic coin of information transfer in the central architecture of this mechanism is composed of one ex- nervous system is the spike: a short electrical pulse that citatory and one inhibitory neuronal populations, with propagates along the axon (output branch) of the trans- reciprocal connections (Fig. 1a). A target rhythm is mitting neuron to synaptic terminals that relay the infor- obtained by tuning the strengths of the excitatory and mation to the dendrites (input branch) of the receiving inhibitory interactions (Fig. 1b-1c). However, it is un- neurons downstream. While spikes are stereotypical, the clear which mechanism results in the required fine-tuning relayed signal depends on the synaptic weight, which can [23, 25]. be thought of as interaction strength. Learning is the We hypothesized that activity dependent synaptic process that modifies synaptic weights (the dynamics of plasticity can provide the mechanism for tuning the inter- the interaction strengths themselves), and typically oc- action strengths in order to stabilize a specific rhythmic curs on a slower timescale than the timescale of the neu- activity in the gamma band. ronal responses. Here we focused on spike timing dependent plastic- STDP is an empirically observed microscopic learning ity (STDP) as the rhythmogenic process [23, 25]. Be- rule in which the modification of the synaptic weight de- low, we briefly describe STDP and derive the dynam- pends on the temporal relation between the spike times ics of the synaptic weights in the limit of slow learning. of the pre (transmitting) and post (receiving) synaptic neurons [26{30]. Following [31] the change, ∆Jij, in the synaptic weight Jij from the pre-synaptic neuron j to the ∗ [email protected] post-synaptic neuron i is expressed as the sum of two pro- bioRxiv preprint doi: https://doi.org/10.1101/2020.07.23.217026; this version posted March 31, 2021. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 2 that learning occurs on a slower timescale than the char- acteristic timescales that describe neuronal activity (for given fixed synaptic weights). A wide range of temporal structures of STDP rules has been reported [32{39]. Here we focus on two families of rules. One is composed of temporally symmetric rules and the other is made up of temporally asymmetric rules. For the temporally symmetric family we apply a dif- ference of Gaussian STDP rule; namely: 1 2 −(t/τ±) =2 K±(t)= p e ; (2) 2πτ± (a) (b) where τ± denotes the characteristic time scales of the potentiation (+) and depression (−). Consistent with the popular description of the famous Hebb rule that `neurons that fire together wire together' [40], we refer to the case of τ+ < τ− as Hebbian, and τ+ < τ− anti- Hebbian. For the temporally a-symmetric STDP rules we take: 1 ∓Ht/τ± K±(t)= e Θ(±Ht); (3) τ± (c) where Θ(x) is the Heaviside step function, τ± are the (d) characteristic time scales of the potentiation (+) and de- pression (−), and H = ±1 dictates the Hebbianity of FIG. 1: The excitatory-inhibitory network. (a) Model the STDP rule. The rule will be termed Hebbian for architecture. The neuronal network here is composed of an H = 1, when potentiation occurs in the causal branch, excitatory (E) and an inhibitory (I) populations with inter- tpost > tpre, and anti-Hebbian for H = −1. (JIE and JEI ) and intra- (JII and JEE , that will be taken Different types of synapses have been reported to ex- to zero hereafter) connections. The interaction is not symmetric and is delayed. (b) The oscillatory dynamics of hibit different types of STDP rules. Consequently, there the mean excitatory end inhibitory population firing rates is no a-priori reason to assume that excitatory and in- mE and mI , respectively, at a frequency of 24:9Hz. Here we hibitory synapses share the exact same learning rule. In used JEE = JII = 0, JIE = 8:91, JEI = 0:9, τm = d = 5ms. particular, the characteristic time constants τE;± for ex- (c) Rhythmic activity. The oscillation frequency is depicted citatory synapses and τI;± for inhibitory synapses may as a function of the strength of the inhibitory to excitatory differ. connection, JEI . The following parameters were Changes to synaptic weights due to the plasticity rule implemented here: JEE = JII = 0, JIE = 8:91, τm = 10ms, of Eq. (1) at short time intervals occur as a result of d = 1ms. (d) Phase diagram of the delayed rate model. either a pre or post-synaptic spike during this interval. Strong inhibition, JI > 1, leads the system to a purely Thus, inhibitory fixed point, in which mE is fully suppressed, and mI = 1. For weak to moderate inhibition, JI 2 (0; 1), and Z 1 _ 0 0 0 0 below the black line, the system converges to a fixed point, Ji;j(t)= λρi(t) ρj(t − t )[K+(t ) − αK−(t )] dt (4) where both populations are active. For J¯ > J¯ (above the 0 d Z 1 black line) the model exhibits rhythmic activity. For a given 0 0 0 0 ¯ ¯ + λρj(t) ρi(t − t )[K+(−t ) − αK−(−t )] dt ; time delay, the frequency is governed by J. When J is 0 increased, higher frequencies are observed. Here we used post=pre τm = d = 1. P where ρpost=pre(t)= l δ(t−tl ) is the spike train of the post/pre neuron written as the sum of the delta func- tion at the neuron's spike times ftpost=preg . In the limit cesses: potentiation (i.e., increasing the synaptic weight) l l of slow learning, λ ! 0, the right hand side of Eq. (4) can and depression (i.e., decreasing the synaptic weight): be replaced by its temporal mean (see [31] for complete ∆Jij = λ [K+(∆t) − αK−(∆t)] ; (1) derivations). This approximation, has been termed the mean field Fokker-Planck approximation [27]. As fluc- where ∆t = ti − tj is the pre and post spike time differ- tuations vanish in this limit and deterministic dynamics ence. Functions K±(t) ≥ 0 describe the temporal struc- are retained for the mean synaptic weights, we get ture of the potentiation (+) and depression (−) of the Z 1 STDP rule. Parameter α denotes the relative strength 0 0 0 0 J_ij(t)= λ Γij(−t )[K+(t ) − αK−(t )] dt ; (5) of the depression and λ is the learning rate.
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