Biological Cybernetics

, Volume 104, Issue 6, pp 369–383 | Cite as

Neuronal model with distributed delay: analysis and simulation study for gamma distribution memory kernel

Original Paper

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 

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References

  1. Baldi P, Atiya AF (1994) How delays affect neural dynamics and learning. IEEE Trans Neural Netw 5: 612–621PubMedCrossRefGoogle Scholar
  2. 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): 155–163PubMedCrossRefGoogle Scholar
  3. Bartholomew DJ (1982) Stochastic models for social processes, 3rd edn. Wiley, LondonGoogle Scholar
  4. Belair J (1993) Stability in a model of a delayed neural network. J Dyn Differ Equ 5: 607–623CrossRefGoogle Scholar
  5. Caianiello ER, De Luca A (1966) Decision equation for binary systems: applications to neuronal behavior. Kybernetik 3: 33–40PubMedCrossRefGoogle Scholar
  6. Chandrasekhar S (1949) Brownian motion, dynamical friction and stellar dynamics. Rev Mod Phys 21(3): 383–388CrossRefGoogle Scholar
  7. 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, MAGoogle Scholar
  8. 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): 802–806PubMedCrossRefGoogle Scholar
  9. De Vries B, Principe JC (1992) The gamma model—a new neural model for temporal processing. Neural Netw 5: 565–576CrossRefGoogle Scholar
  10. 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): 89–110PubMedCrossRefGoogle Scholar
  11. 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, MAGoogle Scholar
  12. Gardiner CW (1986) Handbook of stochastic methods for physics, chemistry and the natural sciences. Springer, New YorkGoogle Scholar
  13. Gerstein GL, Mandelbrot B (1964) Random walk models for the spike activity of a single neuron. Biophys J 4(1, Part 1): 41–68PubMedCrossRefGoogle Scholar
  14. 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): 215–231CrossRefGoogle Scholar
  15. Gopalsamy K, He XZ (1994) Stability in asymmetric Hopfield nets with transmission delays. Physica D 76: 344–358CrossRefGoogle Scholar
  16. Gopalsamy K, Leung IKC (1997) Convergence under dynamical thresholds with delays. IEEE Trans Neural Netw 8(2): 341–348PubMedCrossRefGoogle Scholar
  17. Haken H (1977) Synergetics: introduction and advanced topics. Springer, BerlinGoogle Scholar
  18. Holden AV (1976) Models of the stochastic activity of neurones. Lecture notes in biomathematics. Springer, BerlinGoogle Scholar
  19. 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: 583–594PubMedGoogle Scholar
  20. Izhikevich EM (2001) Resonate-and-fire neurons. Neural Netw 14(6-7): 883–894PubMedCrossRefGoogle Scholar
  21. 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: 2125–2155PubMedCrossRefGoogle Scholar
  22. BS (1977) Brownian motion of a particle with frequency dependent friction. Indian National Science Academy 43: 461–464Google Scholar
  23. BS (2003) Entropy measures, maximum entropy principle and emerging applications. Springer, New YorkGoogle Scholar
  24. Kloeden PE, Platen E (1992) Numerical solution of stochastic differential equations. Springer, BerlinGoogle Scholar
  25. Koch C (1997) Computation and the single neuron. Nature 385 (6613): 207–210PubMedCrossRefGoogle Scholar
  26. Koch C (1999) Biophysics of computation: information processing in single neurons. Oxford University Press, OxfordGoogle Scholar
  27. 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): 63–82PubMedCrossRefGoogle Scholar
  28. Kostal L, Lansky P (2006) Similarity of interspike interval distributions and information gain in a stationary neuronal firing. Biol Cybern 94(2): 157–167PubMedCrossRefGoogle Scholar
  29. Lansky P (1984) On approximation of Steins neuronal model. J Theor Biol 107(4): 631–647PubMedCrossRefGoogle Scholar
  30. Lansky P, Lanska V (1994) First-passage-time problem for simulated stochastic diffusion processes. Comput Biol Med 24(2): 91–101PubMedCrossRefGoogle Scholar
  31. Lansky P, Sanda P, He J (2006) The parameters of the stochastic leaky integrate-and-fire neuronal model. J Comput Neurosci 21(2): 211–223PubMedCrossRefGoogle Scholar
  32. 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): 545–561PubMedCrossRefGoogle Scholar
  33. Llinas RR (1988) The intrinsic electro-physiological properties of mammalian neurons: insights into central nervous system function. Science 242: 1654–1664PubMedCrossRefGoogle Scholar
  34. 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: 897–901PubMedCrossRefGoogle Scholar
  35. MacDonald N (1978) Time lags in biological models. Lecture notes in biomathematics. Springer, BerlinGoogle Scholar
  36. Manwani A, Koch C (1999) Detecting and estimating signals in noisy cable structures, I. Neuronal noise sources. Neural Comput 11(8): 1797–1829PubMedCrossRefGoogle Scholar
  37. 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): 10450–10455PubMedCrossRefGoogle Scholar
  38. 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): 028101PubMedCrossRefGoogle Scholar
  39. Pedroarena C, Llinas RR (1997) Dendritic calcium conductances generate high-frequency oscillation in thalamo-cortical neurons. Proc Natl Acad Sci USA 94: 724–728PubMedCrossRefGoogle Scholar
  40. Ricciardi LM, Sacerdote L (1979) The Ornstein–Uhlenbeck process as a model for neuronal activity. Biol Cybern 35(1): 1–9PubMedCrossRefGoogle Scholar
  41. 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 InstituteGoogle Scholar
  42. Smith H (2011) An introduction to delay differential equations with applications to the life sciences. Texts in applied mathematics, vol 57. Springer, BerlinCrossRefGoogle Scholar
  43. Stein RB (1965) A theoretical analysis of neuronal variability. Biophys J 5(2): 173–194PubMedCrossRefGoogle Scholar
  44. Sullivan WE, Konishi M (1986) Neural map of interaural phase difference in the owl’s brainstem. Proc Natl Acad Sci USA 83(21): 8400–8404PubMedCrossRefGoogle Scholar
  45. 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): 629–637PubMedCrossRefGoogle Scholar
  46. Tank DW, Hopfield JJ (1987) Neural computation by concentrating information in time. Proc Natl Acad Sci USA 84(7): 1896–1900PubMedCrossRefGoogle Scholar
  47. Trivedi KS (2002) Probability and statistics with reliability, queueing and computer science applications, 2nd edn. John Wiley and Sons, New YorkGoogle Scholar
  48. Tuckwell HC, Cannings C, Hoppensteadt FC (1988) Introduction to theoretical neurobiology. Cambridge University Press, Cambridge, CambridgeshireCrossRefGoogle Scholar
  49. 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: 253–271CrossRefGoogle Scholar
  50. Verechtchaguina T, Sokolov IM, Schimansky-Geier L (2006a) First passage time densities in non-Markovian models with subthreshold oscillations. Europhys Lett 73(5): 691–697CrossRefGoogle Scholar
  51. Verechtchaguina T, Sokolov IM, Schimansky-Geier L (2006b) First passage time densities in resonate-and-fire models. Phys Rev E 73(3): 031108CrossRefGoogle Scholar
  52. Ye H, Michel AN, Wing K (1994) Global stability and local stability of Hopfield neural networks with delays. Phys Rev E 50: 4206–4213CrossRefGoogle Scholar
  53. Zwanzig R, Bixon M (1970) Hydrodynamic theory of the velocity correlation function. Phys Rev A 2: 1–8CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Jawaharlal Nehru UniversityNew DelhiIndia
  2. 2.University of Texas at DallasRichardsonUSA

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