LIF Neuron with Hypo-exponential Distributed Delay: Emergence of Unimodal, Bimodal, Multimodal ISI Distribution with Long Tail

Author(s): Saket K. Choudhary, Vijender K. Solanki*

Journal Name: Recent Patents on Engineering

Volume 14 , Issue 2 , 2020

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Graphical Abstract:


Background: Distributed Delay Framework (DDF) has suggested a mechanism to incorporate the delay factor in the evolution of the membrane potential of a neuron model in terms of distributed delay kernel functions. Incorporation of delay in neural networks provide comparatively more efficient output. Depending on the parameter of investigation, there exist a number of choices of delay kernel function for a neuron model.

Objective: We investigate the Leaky integrate-and-fire (LIF) neuron model in DDF with hypoexponential delay kernel. LIF neuron with hypo-exponential distributed delay (LIFH) model is capable to regenerate almost all possible empirically observed spiking patterns.

Methods: In this article, we perform the detailed analytical and simulation based study of the LIFH model. We compute the explicit expressions for the membrane potential and its first two moment viz. mean and variance, in analytical study. Temporal information processing functionality of the LIFH model is investigated during simulation based study.

Results: We find that the LIFH model is capable to reproduce unimodal, bimodal and multimodal inter-spike- interval distributions which are qualitatively similar with the experimentally observed ISI distributions.

Conclusion: We also notice the neurotransmitter imbalance situation, where a noisy neuron exhibits long tail behavior in aforementioned ISI distributions which can be characterized by power law behavior.

Keywords: Fokker planck equation, hypo-exponential distribution, ISI distribution, LIF neuron, laplace transform, multimodal distribution, power law.

S. Dutta, V. Kumar, A. Shukla, N.R. Mohapatraand, and U. Ganguly, "Leaky integrate and fire neuron by charge-discharge dynamics in floating-body MOSFET", Sci. Rep., vol. 07, no. 8257, pp. 1-7, 2017.
[ ] [PMID: 28811481]
S.K. Choudhary, K. Singh, and V.K. Solanki, "Spiking activity of a LIF neuron in distributed delay framework", IJAIM, vol. 3, no. 7, pp. 70-76, 2016.
R.B. Northrop, Ed., Introduction to Dynamic Modeling of Neuro-Sensory Systems., CRC Press, 2000.
B. Kriener, M. Helias, S. Rotter, M. Diesmann, and G.T. Einevoll, "How pattern formation in ring networks of excitatory and inhibitory spiking neurons depends on the input regime", Front. Comput. Neurosci., vol. 7, no. 187, pp. 1-21, 2014.
[ ] [PMID: 24501591]
D.K. Freeman, W.F. Hein, and C.L. Passaglia, "The maintained discharge of rat retinal ganglion cells", Vis. Neurosci., vol. 25, no. 4, pp. 535-548, 2008.
[ ] [PMID: 18634718]
E. Wallance, M. Benayoun, W. Drongelen, and J.D. Cowan, "Emergent oscillations in networks of stochastic spiking neurons", PLoS One, vol. 6, no. 5, 2011.e14804
[ ] [PMID: 21573105]
D.S. Reich, F. Mechler, K.P. Purpura, and J.D. Victor, "Inter spike intervals, receptive fields, and information encoding in primary visual cortex", J. Neurosci., vol. 20, no. 5, pp. 1964-1974, 2000.
[ ] [PMID: 10684897]
R. Sirovivh, L. Sacerdote, and A.E.P. Villa, "Cooperative behavior in a jump diffusion model for a simple network of spiking neurons", Math. Biosci. Eng., vol. 11, no. 4, pp. 385-401, 2014.
[ ] [PMID: 24245723]
T. Hedrick, and J. Waters, "Spiking patterns of neocortical L5 pyramidal neurons in-vitro change with temperature", Front. Cell. Neurosci., vol. 5, no. 1, pp. 1-6, 2011.
[ ] [PMID: 21286222]
M.S.A. Ferraz, H.L.C. Melo-Silva, and A.H. Kihara, "Optimizing information processing in neural networks, beyond critical states", PLoS One, vol. 12, no. 9, 2017.e0184367
[ ] [PMID: 28922366]
S.K. Choudhary, "Emergence of power law behavior in threshold based neuron model with stochastic membrane decay constant", Int. J. App. Math. Inform., vol. 12, pp. 21-28, 2018.
S.K. Sharma, IEEE Trans. Nanobioscience, vol. 12, no. 1, pp. 1-12, 2013.
V.G. Karmeshu, and K.V. Kadambari, "Neuronal model with distributed delay: analysis and simulation study for gamma distribution memory kernel", Biol. Cybern., vol. 104, pp. 369-383, 2011.
[ ] [PMID: 21701877]
P.E. Kloeden, and E. Platen, Numerical Solution of Stochastic Differential Equations., Springer: Berlin, 1992.
S.K. Choudhary, K. Singh, and S.K. Bharti, "Variability in spiking pattern of leaky integrate-and-fire neuron due to excitatory and inhibitory potentials",
M. Orejas, Z. Fedra, and J. Raasakka, Systems and methods for NeQuick modeling using neural networks, .
M. Shoaib, and J. Liu, Hardware efficient deep convolution neural networks, .
B. Kolb, and I.Q. Whishaw, An Introduction to Brain and Behavior., 4th ed Worth Publishers, 2012.
H.C. Tuckwell,
R.L.T. Goris, J.A. Movshon, and E.P. Simoncelli, "Partitioning neuronal variability", Nat. Neurosci., vol. 17, no. 6, pp. 858-867, 2014.
[ ] [PMID: 24777419]
S. Ostojic, "Two types of asynchronous activity in networks of excitatory and inhibitory spiking neurons", Nat. Neurosci., vol. 17, no. 4, pp. 594-602, 2014.
[ ] [PMID: 24561997]
A. Destexhe, and M.R. Lilith, Neuronal Noise.Computational Neuroscience., Springer: USA, 2012.
E.M. Izhikevich, Dynamical Systems in Neuroscience.The Geometry of Excitability and Bursting., MIT Press, 2007.
L.F. Abbott, and P. Dayan, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems., MIT press, 2001.
K. Smaili, T. Kadri, and S. Kadry, "Hypo-exponential distribution with different parameters", Appl. Math., vol. 4, pp. 624-631, 2013.
W. Gerstner, and W.M. Kistler, Spiking Neuron Models: Single Neurons, Populations, Plasticity., Cambridge University Press, 2002.
S.M. Ross, Introduction to Probability Models., 9th ed Academic Press, 2007.
Z. Israel, and K.J. Burcheil, Microelectrode Recording in Movement Disorder Surgery., Thieme Medical Publishers, 2004.
F. Gabbiani, and C. Koch, Principles of Spike Train Analysis.Methods in Neuronal Modeling: From Ions to Networks., MIT Press: Cambridge, 1998.
D.J. Higham, "An algorithmic introduction to numerical simulation of stochastic differential equations", SIAM Rev., vol. 43, no. 3, pp. 525-546, 2001.
S. Kumar, Biosystems, vol. 166, pp. 43-49, 2018.
[ ] [PMID: 29505794]
T.T. Soong, Random Differential Equations in Science and Engineering., Academic Press, Inc., 1973.
H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics., Springer: Berlin, 2011.
S.K. Choudhary, and S.K. Bharti, "Information processing in neuron with exponential distributed delay", Int. J. Machine Learn. Networked Collab. Eng., vol. 2, no. 2, pp. 58-66, 2018.
S. Olmi, D.A. Garcia, A. Imparato, and A. Torcini, "Exact firing time statistics of neurons driven by discrete inhibitory noise", Sci. Rep., vol. 7, no. 1577, pp. 1-15, 2017.
S.K. Sharma, and S. Kumar, IEEE Trans. Nanobioscience, vol. 17, no. 3, pp. 329-341, 2018.
K.J. Miller, L.B. Sorensen, J.G. Ojemann, and M.D. Nijs, Power-law scaling in the brain surface electric potential, .
S.E. Boustani, O. Marre, S. Behuret, P. Baudot, P. Yger, T. Bal, A. Destexhe, and Y. Fregnac, "Network-state modulation of power-law frequency-scaling in visual cortical neurons", PLOS Comput. Biol., vol. 5, no. 9, pp. 1-18, 2009.
[ ] [PMID: 19779556]
T.D. Frank, Nonlinear Fokker-Planck Equations: Fundamentals and Applications., Springer, 2005.

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Article Details

Year: 2020
Published on: 28 October, 2020
Page: [148 - 160]
Pages: 13
DOI: 10.2174/1872212113666190315165139
Price: $25

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