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Numerical study of singularly perturbed differential-difference equation arising in the modeling of neuronal variability. (English) Zbl 1238.65063
Summary: We present a numerical study of a class of boundary value problems of singularly perturbed differential difference equations (SPDDE) which arise in computational neuroscience in particular in the modeling of neuronal variability. The mathematical modeling of the determination of the expected time for the generation of action potential in the nerve cells by random synaptic inputs in dendrites includes a general boundary-value problem for singularly perturbed differential difference equation with shifts. The problem considered in this paper exhibit turning point behavior which add to the complexity in the construction of numerical approximation to the solution of the problem as well as in obtaining theoretical estimates on the solution. Exponentially fitted finite difference scheme based on Il’in-Allen-Southwell fitting is used on a specially designed mesh. Some numerical examples are given to validate convergence and computational efficiency of the proposed numerical scheme. Effect of the shifts on the layer structure is illuminated for the considered examples.

65L03Functional-differential equations (numerical methods)
92C20Neural biology
65L11Singularly perturbed problems for ODE (numerical methods)
34B99Boundary value problems for ODE
39A99Difference equations
Full Text: DOI
[1] Longtin, A.; Milton, J.: Complex oscillations in the human pupil light reflex with mixed and delayed feedback, Math. biosci. 90, 183-199 (1988)
[2] Murray, J. D.: Mathematical biology I: An introduction, (2001)
[3] Mackey, M. C.; Glass, L.: Oscillations and chaos in physiological control systems, Science 197, 287-289 (1977)
[4] Els’gol’ts, E. L.: Qualitative methods in mathematical analysis, Translations of mathematical monographs 12 (1964) · Zbl 0133.37102
[5] Derstein, M. W.; Gibbs, H. M.; Hopf, F. A.; Kaplan, D. L.: Bifurcation gap in a hybrid optical system, Phys. rev. A 26, 3720-3722 (1982)
[6] Segundo, J. P.; Perkel, D. H.; Wyman, H.; Hegstad, H.; Moore, G. P.: Input--output relations in computer simulated nerve cell: influence of the statistical properties, strength, number and inter-dependence of excitatory pre-dependence of excitatory pre-synaptic terminals, Kybernetik 4, 157-171 (1968)
[7] Fienberg, S. E.: Stochastic models for a single neuron firing trains: a survey, Biometrics 30, 399-427 (1974) · Zbl 0286.92003 · doi:10.2307/2529198
[8] Holden, A. V.: Models of the stochastic activity of neurons, (1976) · Zbl 0353.92001
[9] Stein, R. B.: A theoretical analysis of neuronal variability, Biophys. J. 5, 173-194 (1965)
[10] Stein, R. B.: Some models of neuronal variability, Biophys. J. 7, 37-68 (1967)
[11] Johannesma, P. I. M.: Diffusion models of the stochastic activity of neurons, Neural networks: Proceedings of the school on neural networks, 116-144 (1968)
[12] Tuckwell, H. C.: Synaptic transmission in a model for stochastic neural activity, J. theoret. Biol. 77, 65-81 (1979)
[13] Lange, C. G.; Miura, R. M.: Singular perturbation analysis of boundary value problems for differential--difference equations, SIAM J. Appl. math. 42, 502-531 (1982) · Zbl 0515.34058 · doi:10.1137/0142036
[14] Lange, C. G.; Miura, R. M.: Singular perturbation analysis of boundary value problems for differential--difference equations. V. small shifts with layer behavior, SIAM J. Appl. math. 54, 249-272 (1994) · Zbl 0796.34049 · doi:10.1137/S0036139992228120
[15] Lange, C. G.; Miura, R. M.: Singular perturbation analysis of boundary value problems for differential--difference equations. VI. small shifts with rapid oscillations, SIAM J. Appl. math. 54, 273-283 (1994) · Zbl 0796.34050 · doi:10.1137/S0036139992228119
[16] Kadalbajoo, M. K.; Sharma, K. K.: Numerical treatment of a mathematical model arising from a model of neuronal variability, J. math. Anal. appl. 307, 606-627 (2005) · Zbl 1062.92012 · doi:10.1016/j.jmaa.2005.02.014
[17] Kadalbajoo, M. K.; Ramesh, V. P.: Numerical methods on shishkin mesh for singularly perturbed delay differential equations with a grid adaptation strategy, Appl. math. Comput. 188, 1816-1831 (2007) · Zbl 1120.65089 · doi:10.1016/j.amc.2006.11.046
[18] Kadalbajoo, M. K.; Sharma, K. K.: A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations, Appl. math. Comput. 197, 692-707 (2008) · Zbl 1141.65062 · doi:10.1016/j.amc.2007.08.089
[19] Patidar, K. C.; Sharma, K. K.: Uniformly convergent non-standard finite difference methods for singularly perturbed differential--difference equations with delay and advance, Internat. J. Numer. methods engrg. 66, 272-296 (2006) · Zbl 1123.65078 · doi:10.1002/nme.1555
[20] Berger, A. E.; Han, H.; Kellog, R. B.: A priori estimates and analysis of a numerical method for a turning point problem, Math. comp. 42, No. 166, 465-492 (1984) · Zbl 0542.34050 · doi:10.2307/2007596
[21] Farrell, P. A.: Sufficient conditions for the uniform convergence of a difference scheme for a singularly perturbed turning point problem, SIAM J. Numer. anal. 25, No. 3, 618-643 (1988) · Zbl 0646.65068 · doi:10.1137/0725038
[22] Doolan, E. P.; Miller, J. J. H.; Schilders, W. H. A.: Uniform numerical methods for problems with initial and boundary layers, (1980) · Zbl 0459.65058