The firing of an excitable neuron in the presence ofstochastic trains of strong synaptic inputsJonathan RubinDepartment of MathematicsUniversity of PittsburghPittsburgh, PA 15260 USAKreˇsimir Josi´cDepartment of MathematicsUniversity of HoustonHouston, TX 77204-3008, USANovember 9, 2005AbstractWe consider a fast-slow excitable system subject to a stochastic excitatory inputtrain, and show that under general conditions its long term behavior is captured by anirreducible Markov chain. In particular, the firing probability to each input, expectednumber of response failures between firings, and distribution of slow variable valuesbetween firings can be obtained analytically from the distribution of interexcitationintervals. The approach we present immediately generalizes to any pair of input trains,excitatory or inhibitory and synaptic or not, with distinct switching frequencies. Wealso discuss how the method can be extended to other models, such as integrate-and-fire, that feature a single variable that builds up to a threshold where an instantaneousspike and reset occur. The Markov chain analysis guarantees the existence of a limitingdistribution and allows for the identification of different bifurcation events, and thushas clear advantages over direct Monte Carlo simulations. We illustrate this analysis ona model thalamocortical (TC) cell subject to two example distributions of excitatorysynaptic inputs, in the cases of constant and rhythmic inhibition. The analysis showsthat there is a drastic drop in the likelihood of firing just after inhibitory onset in thecase of rhythmic inhibition, relative even to the case of elevated but constant inhibition.This observation provides support for a possible mechanism for the induction of motorsymptoms in Parkinson’s disease, analyzed in [Rubin and Terman, 2004].11 IntroductionThere has been substantial discussion of the roles of excitatory and inhibitory synapticinputs in driving or modulating neuronal firing. Computational analysis of this issue gen-erally considers a neuron awash in a sea of synaptic bombardment [Somers et al., 1998,van Vreeswijk and Sompolinsky, 1998, De Schutter, 1999, Tiesinga et al., 2000, Tiesinga, 2005,Chance et al., 2002, Tiesinga and Sejnowski, 2004, Huertas et al., 2005]. In this work, wealso investigate the impact of synaptic inputs on the firing of a neuron, but with a focus onthe effects of single inputs within stochastic trains. This investigation is motivated by consid-eration of thalamocortical relay (TC) cells, under the hypothesis that such cells are configuredto reliably relay individual excitatory inputs, arising either from strong, isolated synapticsignals or from tightly synchronized sets of synaptic signals, during states of attentive wake-fulness, yet are also modulated by inhibitory input streams [Smith and Sherman, 2002]. Thisviewpoint leads to the question of how the relationship between the activity of a neuron anda stochastic excitatory input train varies under different patterns of inhibitory modulation.The main goal of this paper is to introduce and illustrate a mathematical approach tothe analysis of this relationship. Our approach applies to general excitable systems withseparation of time scales, including a single slow variable, and fast onset and offset of inputs.These ideas directly generalize to other neuronal models, such as integrate-and-fire, featuringa slow build-up of potential interrupted by instantaneous spikes and resets. We harness thesefeatures to reduce system dynamics to a one-dimensional map on the slow variable. Eachiteration of the map corresponds to the time interval from the arrival of one excitatory inputto the arrival of the next excitatory input [Othmer and Watanabe, 1994, Xie et al., 1996,Ichinose et al., 1998, Othmer and Xie, 1999, Coombes and Osbaldestin, 2000]. From thismap, under the assumption of a bounded excitatory input rate, we derive an irreducibleMarkov chain, whose bins are indexed by slow variable values and numbers of inputs re-ceived since the last firing.We prove the key result that under rather general conditions, this Markov chain is ape-riodic, and hence has a limiting distribution. This limiting distribution can be computedfrom the distribution of input arrival times. Once obtained, it can be used to deduce muchabout the firing statistics of the driven cell including the probability that the cell will fire inresponse to a given excitatory input, the expected number of response failures that the cellwill experience between firings, and the distribution of slow variable values attained afterany fixed number of unsuccessful inputs arriving between firings. We emphasize that theguaranteed existence of a limiting distribution constitutes a key advantage of the Markovchain framework over direct Monte Carlo simulations for attaining these types of statistics,since there is no guarantee of convergence for the Monte Carlo approach. Moreover, as weillustrate, the limiting distribution for the Markov chain can be computed analytically, elim-inating the need for simulations altogether. Finally, the Markov chain analysis allows forthe identification of bifurcation events in which variation of model parameters can lead toabrupt changes that affect long-term statistics, although we do not pursue this in detail inthis work (see [Doi et al., 1998, Tateno and Jimbo, 2000], which we also comment upon in2the Discussion in Section 9).We discuss the Markov chain approach in the particular cases of constant inhibition,which may be zero or nonzero, and inhibition that undergoes abrupt switches between twodifferent levels. These choices are motivated by the analysis of TC cell relay reliability inthe face of variations in inhibitory basal ganglia outputs that arise in Parkinson’s disease(PD) and under deep brain stimulation (DBS), applied to combat the motor symptoms ofPD. An important point emerging from experimental results is that parkinsonian changes inthe basal ganglia induce rhythmicity in inhibitory basal ganglia outputs [Nini et al., 1995,Magnin et al., 2000, Raz et al., 2000, Brown et al., 2001], while DBS regularizes these out-puts, albeit at higher than normal levels [Anderson et al., 2003, Hashimoto et al., 2003]. Inrecent work, Rubin and Terman provided computational and analytical support for the hy-pothesis that, given a TC cell that can respond reliably to excitatory inputs under normalconditions, parkinsonian modulations to basal ganglia