Correlations and Synchrony in Threshold Neuron Models

arXiv:0810.2901v3 [q-bio.NC] 17 Jul 2009 Correlations and Synchrony in Threshold Neuron Models Tatjana Tchumatchenko,1, 2 Aleksey Malyshev,3, 4 Theo Geisel,1 Maxim Volgushev,3, 5, 4 and Fred Wolf1 1Max Planck Institute for Dynamics and Self-Organization and Bernstein Center for Computational Neuroscience G¨ottingen, Germany 2G¨ottingen Graduate School for Neurosciences and Molecular Biosciences, Germany 3Inst. of Higher Nervous Activity and Neurophysiology, RAS, Moscow, Russia 4Dep. of Psychology, University of Connecticut, Storrs, USA 5Dep. of Neurophysiology, Ruhr-University Bochum, Germany (Dated: November 4, 2018) We study how threshold model neurons transfer temporal and interneuronal input correlations to correlations of spikes. We find that the low common input regime is governed by firing rate dependent spike correlations which are sensitive to the detailed structure of input correlation functions. In the high common input regime the spike correlations are insensitive to the firing rate and exhibit a universal peak shape independent of input correlations. Rate heterogeneous pairs driven by common inputs in general exhibit asymmetric spike correlations. PACS numbers: 87.19.lm, 87.19.ll, 05.40.-a, 87.19.lt, 87.85.dm Neurons in the CNS exhibit temporally correlated ac- tivity that can reflect specific features of sensory stim- uli or behavioral tasks [1, 3]. Recently, the origin, sta- tistical structure and coding properties of spike corre- lations in neuronal systems have attracted substantial attention [2, 4, 5]. How do neurons transfer correlated inputs into correlated output? This fundamental ques- tion is vital to understand the structure of network cor- relations, yet it is unanswered. In the past, most theo- retical analyses addressing this question utilized coupled stochastic differential equations and the Fokker Planck formalism [5, 7]. These approaches are technically very demanding and are therefore practically restricted to sim- ple stochastic processes, see e.g. [7]. As a result, explicit expressions for quantities of interest are often lacking or obtainable only in special limiting cases. Here we show that an alternative modeling framework, based on the theory of smooth random functions [8, 9], can provide a mathematically transparent and highly tractable description of spike correlations driven by in- puts of arbitrary temporal structure and correlation strength. Our theoretical findings may find applications beyond neuroscience, e.g. in spin ordering, reliability studies, as the statistics of (upward) level crossings is a general, long standing problem in physics and engi- neering [9]. We calculate quantities which have so far escaped a theoretical description by competing Fokker- Planck based formalism: 1) peak spike correlation for arbitrary input correlation strength, 2) rate independent peak shape in the high correlation regime, 3) complete spike correlation function for weak correlations, 4) asym- metric spike correlation function in rate inhomogeneous pairs. A priori, the simple threshold model we use cannot be expected to completely capture the complex behavior of cortical neurons. Remarkably, our results reproduce and extend previous reports on the firing rate depen- dence of cortical spike correlations [4, 5]. Furthermore, −60 −40 −20 V in mV

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r≈1 r≈0.3 B FIG. 1: (A) Paradigm of two fluctuating current traces which share a component nC and have a correlation strength r(r > 0). The correlated currents are injected successively into neurons. (B) MP traces of two neurons (red and blue) recorded in response to correlated fluctuating current with r ≈1 and r ≈0.3. In all four recordings ν ≈5Hz, τs=20ms and CI(τ) as in Eqs. 1,4; spikes are truncated at −10mV we were able to qualitatively confirm all new predictions in our in vitro experiments. Framework– We model the membrane potential (MP) of a neuron by a temporally continuous, stationary Gaus- sian random function V (t) with temporal correlation C(τ)=⟨V (0)V (τ)⟩and zero mean, which we term Gaus- sian Pseudo Potentials (GPPs); ⟨·⟩denotes the ensemble average. Assuming a simple leaky integrator model with a membrane time constant τM, τM ˙V (t)=−V (t) + ξ(t). The voltage and current statistics are linked by: eCI(ω) = eC(ω)(1 + τ 2 Mω2). (1) where CI(τ) denotes the current correlation function and ˜CI(ω) its Fourier transform. To induce correla- tions we use common input. Two correlated currents ξj(t), j = 1, 2 (1 and 2 denote noise input to neurons 1 and 2) are derived from three statistically independent temporally correlated Gaussian processes n1, n2, nc with current correlation function CI(τ): ξj(t) = √ 1 −rnj(t) + √rnc(t). (2) nj are the individual noise components (as in Fig. 1 (A)) and nc the shared component. Vj and nj can be related 2 using the membrane filter in Eq. 1. The corresponding correlated voltages V1, V2 mimic the neuronal traces of two neurons subject to common input. r modulates be- tween fu

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