Phase-locking and bistability in neuronal networks with synaptic depression
Introduction
Coherent activity in deterministic networks of coupled oscillators often takes the form of phase-locked activity. In this situation, relative to some common reference point, each network element is assigned a phase that is periodic over time. The relative phase differences between the network elements can then be computed to determine potential phase-locked states. Such networks arise in a variety of physical and biological contexts, such as cardiac networks [1], central pattern generating neuronal networks [2], and those described by weakly-coupled Kuramoto oscillators [3].
Various mathematical approaches have been developed to understand phase-locking. One of the most common methods relies on weak coupling among the network elements, so that the technique of averaging can be applied. This allows the phase relationship between the network elements to be systematically reduced to the study of sets of equations on a torus, whose roots correspond to phase-locked states [4]. Along similar lines, phase models have been used in large networks of globally coupled oscillators to derive a continuum description of phases where the existence and stability of clustered, incoherent or synchronized states is studied [[5], [6]]. Another common method uses the phase response curve (PRC) to derive maps whose fixed points correspond to phase-locked solutions [7]. The PRC measures the response of an oscillator to perturbations given at specific phases of the oscillation cycle. The PRC is a mapping with domain given by the perturbation phase and range equal to the change of phase of the periodic trajectory. A positive (negative) value of the PRC implies that a perturbation given at that phase causes the oscillator to increase (decrease) its phase relative to a specified reference point. Neuronal models for which the PRC is strictly of one sign are Type I, while those in which the PRC changes sign are Type II [8].
Oscillators may also be subject to inputs that are not necessarily weak. In this case, the spike-time response curve characterizes how the timing of the next spike is affected by an input. By normalizing against the intrinsic period of the neuron, one effectively obtains a phase response curve, albeit one that may not quantitatively match the one obtained from weak perturbations. A synaptic current from a presynaptic neuron can be thought of as an (not necessarily weak) input to a postsynaptic cell that may affect its phase. While there are a wide variety of synapses, we will focus on inhibitory synapses that exhibit short-term synaptic depression where the strength of the synapse increases as a function of period of the presynaptic neuron. We are interested in finding situations where more than one stable periodic solution exists as a result of the short-term synaptic depression.
Multistability of solutions refers to the existence of multiple stable solutions for the same set of parameters. Each of these solutions has a basin of attraction defined as the set of initial conditions for which the starting trajectory asymptotically approaches this solution. Multistability is thought to be of importance to a neuronal network in that each of the stable solutions corresponds to a different network output state. Thus, the capabilities of a network are expanded in the presence of multistability. It has been shown previously that synaptic depression can lead to bistable states in neuronal networks [[9], [10], [11]]. Synaptic depression can enhance information about stimuli in competitive networks that display a multitude of dominance times [12], but can also detract from multistability of dominance times in noise induced switching in excitatory networks [13].
In this study, we show that bistability of different phase-locked states can arise in a pair of Type I neurons in which just one of the synapses exhibits short-term depression. Further, we develop a technique for finding the phase-locked states that relies on knowing only the PRC of each neuron, rather than the specific mathematical equations needed to describe the evolution of a model’s voltage variable. Calculating a PRC of a neuron is a feed-forward process in that the timing of the perturbation to a neuron can be externally controlled. There is significant work on approximating PRCs from experimental data; for example see [[14], [15]]. The maximal synaptic strength as a function of cycle period or frequency (synaptic plasticity profile) can also be calculated in a feed-forward manner [16]. Huang [17] developed a method to combine these two types of feed-forward information into a feedback Poincaré map. The stable (unstable) fixed points of this map corresponded to stable (unstable) phase-locked solutions of the reciprocally coupled inhibitory system. Using Huang’s method, we derive two distinct 2-D maps. For each of these maps, we derive conditions for the existence of bistable solutions. Our analysis reveals that bistability occurs when either the PRC of the neuron or the synaptic plasticity profile of the synapse has a sufficiently steep derivative in a neighborhood of a fixed point. To illustrate our proposed methods, we use the Quadratic Integrate-and-fire (QIF) model [18] and the Morris–Lecar (ML) model [19]. The QIF is the normal form of saddle–node bifurcation of fixed points. From it, one can derive the theta model which is the canonical Type I phase model. We use the QIF model because we can analytically derive its PRC. The ML model is perhaps the most basic, biophysically based planar model of a neuron and is widely used in mathematical and computational studies.
This paper is organized as follows. In Section 2, we describe the coupled systems governed by either the QIF or ML models, together with their respective PRCs. In Section 3 we first derive three distinct maps. The first map is 1-D, previously derived in Dror et al. [7], that describes the behavior of two coupled neurons in which the synapses are static (not depressing). The second two maps are the aforementioned 2-D maps. We show that a stable fixed point of the 1-D map has a corresponding fixed point of either of the 2-D maps, however its stability may be different. In this section, we also utilize a geometric method, developed in [20], to determine existence of bistable solutions. Section 4 concludes with a Discussion.
Section snippets
Models and methods
The main results of this paper hold for neuronal models that display Type I dynamics as described below. We will analytically (numerically) calculate a family of PRCs for the QIF (ML) models. We will use this family of PRCs to construct a 2-D map that determines the existence and stability of phase locked solutions of a reciprocally coupled set of two inhibitory neurons. We will also use the model equations to conduct simulations and show that the results agree with those obtained from the 2-D
Derivation of the maps
For completeness, we start with the derivation of the Poincaré map for the relative firing times of the neurons when they are connected with static synapses [[7], [20]]. Rewriting the PRC relationship (3), we can obtain the cycle lengths of each cell in cycle as Note that we use the PRC where the value in the second argument is chosen for the non-depressing, static case. The following equations relate the firing times of the two cells
Discussion
Numerous theoretical and computational studies have utilized PRCs to explore phase-locking in oscillatory neuronal networks [[7], [18], [20], [26], [27], [28], [29], [30]]; see [31] for a review. Some studies assume short or weak perturbations and use iPRCs [[27], [28]], while others use more general PRCs [[29], [32]] obtained from inputs that are not necessarily weak. In the case of strong inputs, PRCs do not necessarily scale linearly with input strength. Additionally, experimental work has
Acknowledgments
This work was supported, in part, by PSC CUNY 68127-00 46 (ZA), NIH MH060605 (FN) and NSF DMS1122291 (AB).
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