Journal of Theoretical Biology 272 (2011) 42–54 Contents lists available at ScienceDirect Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi Co-existent activity patterns in inhibitory neuronal networks with short-term synaptic depression Amitabha Bose a,c,Ã, Victoria Booth b a Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA b Departments of Mathematics and Anesthesiology, University of Michigan, Ann Arbor, MI 48109, USA c School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India article info abstract Article history: A network of two neurons mutually coupled through inhibitory synapses that display short-term synaptic Received 19 May 2010 depression is considered. We show that synaptic depression expands the number of possible activity Received in revised form patterns that the network can display and allows for co-existence of different patterns. Specifically, the 18 October 2010 network supports different types of n–m anti-phase firing patterns, where one neuron fires n spikes Accepted 1 December 2010 followed by the other neuron firing m spikes. When maximal synaptic conductances are identical, n–n Available online 9 December 2010 anti-phase firing patterns are obtained and there are conductance intervals over which different pairs of Keywords: these solutions co-exist. The multitude of n–m anti-phase patterns and their co-existence are not found Dynamical systems when the synapses are non-depressing. Geometric singular perturbation methods for dynamical systems Periodic orbit are applied to the original eight-dimensional model system to derive a set of one-dimensional conditions Synaptic depression for the existence and co-existence of different anti-phase solutions. The generality and validity of these conditions are demonstrated through numerical simulations utilizing the Hodgkin–Huxley and Morris– Lecar neuronal models. & 2010 Elsevier Ltd. All rights reserved. 1. Introduction 2007) and introducing co-existing stable firing patterns (Bose et al., 2001; Chandrasekaran et al., 2009; Jalil et al., 2004; Manor and Short-term synaptic depression is widely found in synapses of Nadim, 2001; Nadim et al., 1999). Understanding the contributions the central nervous system (Abbott and Regehr, 2004) and in central of synaptic depression to the generation of such diverse patterns of pattern generating networks (O’Donovan et al., 1998; Rabbah and activity is often best achieved by studying small, idealized networks Nadim, 2007). Depressing synapses display changes in their strength of model neurons. Indeed, the study of small minimal models (as as a function of firing frequency. In particular, when the activation small as two neurons) continues to reveal important insights into the frequency of a pre-synaptic cell is high, then its synapse depresses operationofneuronalnetworksofanyscale(Skinner et al., 2005). with each subsequent spike weakening its effect on a post-synaptic Despite their small size, minimal networks display remarkable cell. Alternatively, when the activation frequency is low, the synapse complexity in their dynamic states. recovers to a strong state between each spike and has a much larger In this study, we consider how synaptic depression can be effect on the post-synaptic cell. The effect of synaptic depression on utilized to create co-existence of different types of anti-phase network behavior depends, in some measure, on network structure, solutions in a network of two mutually coupled inhibitory neurons. particularly whether a network is generally feedforward in design or We show that, largely independent of the specific neuron model includes feedback projections among its constituent neurons (see used, depression introduces a multitude of co-existent n–n anti- Grande and Spain, 2005 for a review). In reciprocally connected phase firing states. In an n–n firing pattern, neuron 1 fires n networks, depression has been shown to broaden network dynamics consecutive spikes followed by n consecutive spikes of neuron 2 in a variety of ways, such as by generating rhythmic spontaneous and so on. As the maximal synaptic conductance is increased, the activity (Rubin et al., 2009; Tabak et al., 2000), regulating the relative network can switch from 1–1 to 2–2 to 3–3 and higher anti-phase timing of firing of network components (Manor et al., 2003; Mouser firing. We also find that there are ranges of the maximal synaptic et al., 2008), supporting persistent activity (Barbieri and Brunel, conductance over which pairs of these solutions co-exist. Inter- estingly, when the synapses do not display synaptic depression, these intervals of co-existence largely disappear and the network à Corresponding author at: Department of Mathematical Sciences, New Jersey supports a smaller number of different n–n firing patterns. Institute of Technology, Newark, NJ 07102, USA. Tel.: +1 973 596 3370; fax: +1 973 596 5591. In the context of network bistability with depressing synapses, E-mail address: [email protected] (A. Bose). Nadim et al. (1999) introduced the concept of a cell-controlled 0022-5193/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2010.12.001 A. Bose, V. Booth / Journal of Theoretical Biology 272 (2011) 42–54 43 versus synapse-controlled network. In the former, the intrinsic order equations in which calcium, potassium and leak currents dynamics of the individual neurons determine the network fre- contribute to firing solutions. See Appendix for specific equations quency, which is necessarily high so that the synapses can remain and parameter values. depressed and largely irrelevant. In the synapse-controlled case, The mathematical analysis is performed on a generic relaxation the dynamics of the synapse determine the network frequency oscillator model whose basic dynamics replicate the tonic spiking which is typically much lower than in the cell-controlled scenario. behavior of either of the models that we simulate. We first consider In this paper, we expand on this idea to show that there are several the case when both neurons are identical, intrinsic oscillators with levels of synapse-controlled oscillations each of which corresponds identical synapses. Equations for each neuron can be written as to a different n–n anti-phase firing pattern. Each of these patterns is ev u ¼ f ðv ,w Þgs ½v ÀE , characterized by distinct levels of synaptic depression and recov- i i i j i inh ery. The primary implication of this finding is therefore that the w u ¼ hðv ,w Þ, type of co-existence that we find is largely independent of the i i i intrinsic dynamics of the neurons. As a result, different types of ( ð1ÀdiÞ=ta if vi ovy, neuronal models should be able to exhibit co-existence simply by d u ¼ i Àd =t if v 4v , allowing the synapses between cells to display the appropriate i b i y level of synaptic depression. We illustrate the generality of this ( result through simulations using two different neuron models, the Àsi=tk if vi ovy, siu ¼ ð1Þ Hodgkin–Huxley model and the Morris–Lecar model. We have 0ifvi 4vy, chosen parameters such that the models operate in very different frequency regimes and show how the ensuing network dynamics where i, j¼1, 2. The function f(v,w) incorporates the ionic currents can be understood through an analysis of the synapses between intrinsic to each cell, h(v,w) models the gating of one of the currents, neurons that largely ignores the intrinsic dynamics of these sj is the synaptic gating variable governing the input from neuron j to neurons. neuron i,andvy is the synaptic threshold. The parameters g and Einh A primary goal of this paper is not just to show co-existence of are the maximal conductance and reversal potential of the synapse, solutions, but also to describe a methodology by which properties of respectively. As modeled in Bose et al. (2001),thevariabledi is a thesesolutionscanbeunderstood.Tothatend,weuseacombination depression variable which tracks the extent of depression and of numerical simulations and analysis. In particular, simulations of recovery of the synapse from cell i to cell j and depends on the different neuron models are used to show co-existence of various n–n amount of time a neuron spends above and below threshold. The solutions. Rather than rigorously proving the existence and stability of synaptic gating variable si and the depression variable di are linked these solutions, we use analytic methods to extract the key reasons through the following reset condition: whenever the voltage of why these solutions should exist. One advantage of this approach is neuron i increases above the synaptic threshold vy, si is instanta- that it allows us to isolate the mechanisms that underlie the existence neously reset to the current value of di. More specifically, if a + of these solutions and the transitions between them without becoming presynaptic action potential occurs at t¼0, then we reset si(0 )to too deeply ensnared in the mathematical content. The main mathe- equal di(0). At all other times, except the moment when the matical tool to conduct the analysis is geometric singular perturbation presynaptic neuron increases above threshold, di and si are theory which is used to derive reduced sets of equations that govern the uncoupled and obey Eqs. (1). This discontinuous reset is consistent behavior of solutions along different parts of the trajectory. We focus on with other phenomenological models of synaptic depression (Abbott deriving necessary and sufficient conditions for the existence and et al., 1997; Tsodyks and Markram, 1997). To model a non-depres- co-existence of solutions. In particular, we are able to collapse the sing synapse, we replace the diu equation with di 1. In the non- 8-dimensional network model to the study of two distinct 1-dimen- depressing case, the synaptic variable si is always reset to the value 1 sional objects to establish these conditions.