Neuronal model with distributed delay: analysis and simulation study for gamma distribution memory kernel Authors Karmeshu Jawaharlal Nehru University Varun Gupta University of Texas at Dallas K. V. Kadambari Jawaharlal Nehru University Original Paper

First Online: 24 June 2011 Received: 16 May 2010 Accepted: 30 May 2011 DOI :
10.1007/s00422-011-0441-y

Cite this article as: Karmeshu, Gupta, V. & Kadambari, K.V. Biol Cybern (2011) 104: 369. doi:10.1007/s00422-011-0441-y
Abstract
A single neuronal model incorporating distributed delay (memory)is proposed. The stochastic model has been formulated as a Stochastic Integro-Differential Equation (SIDE) which results in the underlying process being non-Markovian. A detailed analysis of the model when the distributed delay kernel has exponential form (weak delay) has been carried out. The selection of exponential kernel has enabled the transformation of the non-Markovian model to a Markovian model in an extended state space. For the study of First Passage Time (FPT) with exponential delay kernel, the model has been transformed to a system of coupled Stochastic Differential Equations (SDEs) in two-dimensional state space. Simulation studies of the SDEs provide insight into the effect of weak delay kernel on the Inter-Spike Interval(ISI) distribution. A measure based on Jensen–Shannon divergence is proposed which can be used to make a choice between two competing models viz. distributed delay model vis-á-vis LIF model. An interesting feature of the model is that the behavior of (CV(t ))_{(ISI)} (Coefficient of Variation) of the ISI distribution with respect to memory kernel time constant parameter η reveals that neuron can switch from a bursting state to non-bursting state as the noise intensity parameter changes. The membrane potential exhibits decaying auto-correlation structure with or without damped oscillatory behavior depending on the choice of parameters. This behavior is in agreement with empirically observed pattern of spike count in a fixed time window. The power spectral density derived from the auto-correlation function is found to exhibit single and double peaks. The model is also examined for the case of strong delay with memory kernel having the form of Gamma distribution. In contrast to fast decay of damped oscillations of the ISI distribution for the model with weak delay kernel, the decay of damped oscillations is found to be slower for the model with strong delay kernel.

Keywords
Gamma distribution memory kernel
Weak and strong delay
Exponential distributed delay
First passage time
ISI distribution
Coefficient of variation
Autocorrelation function
Power spectral density
Jensen–Shannon divergence

References Baldi P, Atiya AF (1994) How delays affect neural dynamics and learning. IEEE Trans Neural Netw 5: 612–621

PubMed CrossRef Bar-Gad I, Ritov Y, Bergman H (2001) The neuronal refractory period causes a short-term peak in the autocorrelation function. J Neurosci Methods 104(2): 155–163

PubMed CrossRef Bartholomew DJ (1982) Stochastic models for social processes, 3rd edn. Wiley, London

Belair J (1993) Stability in a model of a delayed neural network. J Dyn Differ Equ 5: 607–623

CrossRef Caianiello ER, De Luca A (1966) Decision equation for binary systems: applications to neuronal behavior. Kybernetik 3: 33–40

PubMed CrossRef Chandrasekhar S (1949) Brownian motion, dynamical friction and stellar dynamics. Rev Mod Phys 21(3): 383–388

CrossRef Dayan P, Abbott LF (2003) Theoretical neuroscience: computational and mathematical modeling of neural systems. Published by the MIT Press with the Cognitive Neuroscience Institute, Cambridge, MA

De la Rocha J, Doiron B, Shea-Brown E, Josic K, Reyes A (2007) Correlation between neural spike trains increases with firing rate. Nature 448(7155): 802–806

PubMed CrossRef De Vries B, Principe JC (1992) The gamma model—a new neural model for temporal processing. Neural Netw 5: 565–576

CrossRef Erchova I, Kreck G, Heinemann U, Herz AVM (2004) Dynamics of rat entorhinal cortex layer II and III cells: characteristics of membrane potential resonance at rest predict oscillation properties near threshold. J Physiol 560(1): 89–110

PubMed CrossRef Gabbiani F, Koch C (1998) Principles of spike train analysis. In: Koch C, Segev I (eds) Methods in neuronal modeling: from ions to networks. MIT Press, Cambridge, MA

Gardiner CW (1986) Handbook of stochastic methods for physics, chemistry and the natural sciences. Springer, New York

Gerstein GL, Mandelbrot B (1964) Random walk models for the spike activity of a single neuron. Biophys J 4(1, Part 1): 41–68

PubMed CrossRef Giraudo MT, Sacerdote L, Zucca C (2001) A monte carlo method for the simulation of first passage times of diffusion processes. Methodol Comput Appl Probab 3(2): 215–231

CrossRef Gopalsamy K, He XZ (1994) Stability in asymmetric Hopfield nets with transmission delays. Physica D 76: 344–358

CrossRef Gopalsamy K, Leung IKC (1997) Convergence under dynamical thresholds with delays. IEEE Trans Neural Netw 8(2): 341–348

PubMed CrossRef Haken H (1977) Synergetics: introduction and advanced topics. Springer, Berlin

Holden AV (1976) Models of the stochastic activity of neurones. Lecture notes in biomathematics. Springer, Berlin

Hutcheon B, Miura RM, Yarom Y, Puil E (1994) Low-threshold calcium current and resonance in thalamic neurons: a model of frequency preference. J Neurophysiol 71: 583–594

PubMed Izhikevich EM (2001) Resonate-and-fire neurons. Neural Netw 14(6-7): 883–894

PubMed CrossRef Jackson BS (2004) Including long-range dependence in integrate and fire models of the high interspike interval variability of the cortical neurons. Neural Comput 16: 2125–2155

PubMed CrossRef BS (1977) Brownian motion of a particle with frequency dependent friction. Indian National Science Academy 43: 461–464

BS (2003) Entropy measures, maximum entropy principle and emerging applications. Springer, New York

Kloeden PE, Platen E (1992) Numerical solution of stochastic differential equations. Springer, Berlin

Koch C (1997) Computation and the single neuron. Nature 385 (6613): 207–210

PubMed CrossRef Koch C (1999) Biophysics of computation: information processing in single neurons. Oxford University Press, Oxford

Koch C, Bernander O, Douglas RJ (1995) Do neurons have a voltage or a current threshold for action potential initiation?. J Comput Neurosci 2(1): 63–82

PubMed CrossRef Kostal L, Lansky P (2006) Similarity of interspike interval distributions and information gain in a stationary neuronal firing. Biol Cybern 94(2): 157–167

PubMed CrossRef Lansky P (1984) On approximation of Steins neuronal model. J Theor Biol 107(4): 631–647

PubMed CrossRef Lansky P, Lanska V (1994) First-passage-time problem for simulated stochastic diffusion processes. Comput Biol Med 24(2): 91–101

PubMed CrossRef Lansky P, Sanda P, He J (2006) The parameters of the stochastic leaky integrate-and-fire neuronal model. J Comput Neurosci 21(2): 211–223

PubMed CrossRef Liao X, Li S, Chen G (2004) Bifurcation analysis on a two-neuron system with distributed delays in the frequency domain. Neural Netw 17(4): 545–561

PubMed CrossRef Llinas RR (1988) The intrinsic electro-physiological properties of mammalian neurons: insights into central nervous system function. Science 242: 1654–1664

PubMed CrossRef Llinas RR, Grace AA, Yarom Y (1991) In-vitro neurons in mammalian cortical layer 4 exhibit intrinsic oscillatory in the 10-to-50Hz frequency range. Proc Natl Acad Sci USA 88: 897–901

PubMed CrossRef MacDonald N (1978) Time lags in biological models. Lecture notes in biomathematics. Springer, Berlin

Manwani A, Koch C (1999) Detecting and estimating signals in noisy cable structures, I. Neuronal noise sources. Neural Comput 11(8): 1797–1829

PubMed CrossRef Mar DJ, Chow CC, Gerstner W, Adams RW, Collins JJ (1999) Noise shaping in populations of coupled model neurons. Proc Natl Acad Sci USA 96(18): 10450–10455

PubMed CrossRef Moreno-Bote R, Parga N (2006) Auto-and crosscorrelograms for the spike response of leaky integrate-and-fire neurons with slow synapses. Phys Rev Lett 96(2): 028101

PubMed CrossRef Pedroarena C, Llinas RR (1997) Dendritic calcium conductances generate high-frequency oscillation in thalamo-cortical neurons. Proc Natl Acad Sci USA 94: 724–728

PubMed CrossRef Ricciardi LM, Sacerdote L (1979) The Ornstein–Uhlenbeck process as a model for neuronal activity. Biol Cybern 35(1): 1–9

PubMed CrossRef Ruan S (2004) Delay differential equatioms in single species dynamics. In: Ait Dads E, Arino O, Hbid M (eds) Delay differential equations with applications. NATO Advanced Study Institute

Smith H (2011) An introduction to delay differential equations with applications to the life sciences. Texts in applied mathematics, vol 57. Springer, Berlin

CrossRef Stein RB (1965) A theoretical analysis of neuronal variability. Biophys J 5(2): 173–194

PubMed CrossRef Sullivan WE, Konishi M (1986) Neural map of interaural phase difference in the owl’s brainstem. Proc Natl Acad Sci USA 83(21): 8400–8404

PubMed CrossRef Svirskis G, Rinzel J (2000) Influence of temporal correlation of synaptic input on the rate and variability of firing in neurons. Biophys J 79(2): 629–637

PubMed CrossRef Tank DW, Hopfield JJ (1987) Neural computation by concentrating information in time. Proc Natl Acad Sci USA 84(7): 1896–1900

PubMed CrossRef Trivedi KS (2002) Probability and statistics with reliability, queueing and computer science applications, 2nd edn. John Wiley and Sons, New York

Tuckwell HC, Cannings C, Hoppensteadt FC (1988) Introduction to theoretical neurobiology. Cambridge University Press, Cambridge, Cambridgeshire

CrossRef Turcott RG, Barker PDR, Teich MC (1995) Long-duration correlation in the sequence of action potentials in an insect visual interneuron. J Stat Comput Simul 52: 253–271

CrossRef Verechtchaguina T, Sokolov IM, Schimansky-Geier L (2006a) First passage time densities in non-Markovian models with subthreshold oscillations. Europhys Lett 73(5): 691–697

CrossRef Verechtchaguina T, Sokolov IM, Schimansky-Geier L (2006b) First passage time densities in resonate-and-fire models. Phys Rev E 73(3): 031108

CrossRef Ye H, Michel AN, Wing K (1994) Global stability and local stability of Hopfield neural networks with delays. Phys Rev E 50: 4206–4213

CrossRef Zwanzig R, Bixon M (1970) Hydrodynamic theory of the velocity correlation function. Phys Rev A 2: 1–8

CrossRef