Phase Response Curve of Spike Response Model

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1 Vol.29-MPS-76 No.39 Vol.29-BIO-19 No.39 29/12/18 1 1, 2 3, 2 1, 2 Phase Response Curve of Spike Response Model Munenori Iida, 1 Toshiaki Omori, 1, 2 Toru Aonishi 3, 2 and Masato Okada 1, 2 Phase response curve (PRC) is one of useful tools to understand of population dynamics and has been measured in recent physiological experiments. However, the relation between subthreshold membrane property of a single neuron and its PRC is unclear. In this paper, we derive the PRCs of single neurons by using spike response model and analytically clarify the relation between the subthreshold membrane property and the PRC. We analytically show that the PRC is described by the kernels of the spike response model and verify our theory by numerical simulations. 1 Graduate School of Frontier Sciences, The University of Tokyo 2 RIKEN Brain Science Institute 3 Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology 1. 1),2) 1),2) 3),4) 5) 8) 2 leaky integrate-and-fire model 5),6) 7),8) Gerstner spike response model 9) 11) spike response model 9) 11) 1 c 29 Information Processing Society of Japan

2 t u(t) u(t) = η(t t f ) + κ(s)i ext (t s)ds. (1) u(t) u(t) t f (1) u(t) η(t t f ) η(t t f ) = η exp ( t tf τ η ) Θ(t t f ). (2) 1A η(t t f ) t f 1A η η(t t f ) τ η 2 t s I ext (t s) κ(s) I ext (t s)ds κ(s) κ(s) 1B κ(s) 2 2 κ(s) κ(s) = s cos(ws) exp ( s ) Θ(s), (3) τ s τ s w 1B (3) τ s = 1. w =. 1B w = ( ) κ(s) type1 1B (3) τ s = 1. w = 1. 1B κ(s) s w ( ) κ(s) type2 κ(s) w I ext (t) I ext (t) = I + ϵδ(t t ), (4) δ(x) (4) I ϵδ(t t ) t ϵ η(t-t f ) η t f 1 Fig. 1 threshold resting value t κ(s) s η(t t f )(A) κ(s)(b). The kernels η(t t f )(A) and κ(s)(b) in the spike response model. η(t t f ) κ(s) 2.2 phase response curve 1),2) mV 35mV w = η = 55. τ η = 1.5 τ s =.2 I =.35mA 2 I ext (t) ϵ= I 2 T Vol.29-MPS-76 No.39 Vol.29-BIO-19 No.39 29/12/18 2 c 29 Information Processing Society of Japan

3 Vol.29-MPS-76 No.39 Vol.29-BIO-19 No.39 29/12/18 Fig. 2 membarane potential[mv] current[mv] T T 1 (t ) time[msec] time[msec] ΔT(t ) 2 Periodic activity of the membrane potential in spike response model, and the time shift of the spike time in response to the perturbation stimulus. The time course of membrane potential without perturbation is shown by the solid line and the time course with perturbation is shown by the dotted line. 3 Potential [mv] u(t) u (t) εδ(t-t ) t T- T(t ) T(t ) T u (T) =threshold t [ms] T 1 u (t)( ) t u(t)( ) T (t ) Fig. 3 Schematic diagram used to derive the PRC of the spike response model theoretically. The time course of membrane potential without perturbation is shown by u (t)(solid line), and the time course of membrane potential induced by perturbation at time t is shown by u (t)(dotted line). The dashed-dotted line denotes the firing threshold. T and T (t ) denote the period and the spike time shift, respectively. T t (t (, T ]) 2 2 T 2 T 1(t ) T 1(t ) t T (t ) T 1 (t ) T T (t ) = T T 1(t ), (5) T T 1 (t ) < T T < T 1 (t ) t T t T (t ) t Z(t ) t T t T (t ) Z(t ) Z(t ) = T (t ), (6) ϵ (ϵ=) u (t) (ϵ=) I ext (t) I u (t) (1) u (t) = η(t t f ) + I κ(s)ds. (7) 3 c 29 Information Processing Society of Japan

4 u (t) T u (T ) t 3 u(t) T T (t ) t = T T (t ) T T (t ) ( ϵ) ϵ (1) (4) ϵκ(t t ) T T (t ) ϵκ(t T (t ) t ) T T (t ) u (T T (t )) T T (t ) u (T T (t )) + ϵκ(t T (t ) t ) u (T ) u (T ) = u (T T (t )) + ϵκ(t T (t ) t ). (8) (8) 1 2 ( ) d u (T T (t )) = u (T ) dt u(t ) T (t ) +, (9) ( ) d κ(t T (t ) t ) = κ(t t ) ds κ(t t ) T (t ) +. (1) ( ) d u (T ) u (T ) dt u (T ) T (t ) + { ( ) } d +ϵ κ(t t ) ds κ(t t ) T (t ) +. (11) (6) T (t ) = Z(t )ϵ (11) ϵ ( ) d κ(t t ) dt u (T ) Z(t ) + O(ϵ) =, (12) O(ϵ) ϵ 1 (12) Z(t ) Z(t ) = κ(t t ) ( d dt u (T )) 1. (13) 2.4 (2) (3) η(t t f ) κ(s) (1) u(t) = η e (t tf )/τ η + s cos(ws) exp( s/τ s )I ext (t s)ds, (14) I ext (t s) (4) (ϵ = ) u (t) u (t) = η e (t tf )/τ η + (1 τ 2 s w 2 )τ 2 s (1 + τ 2 s w 2 ) 2 I. (15) T 2 (ϵ = ) T t f t f = u (T ) u thre u rest (15) u thre = u rest η e T /τ η + (1 τ s 2 w 2 )τs 2 I. (16) (1 + τs 2 w 2 ) 2 (16) T { ( 1 + τ 2 s w 2) 2 η T = τ η log (1 + τ 2 s w 2 ) 2 (u rest u thre ) + (1 τ 2 s w 2 ) τ 2 s I }. (17) (13) Z(t ) { } Z(t ) = τ { } η τs τ η t exp T (T t ) cos {w(t t )} exp. (18) η τ s τ η τ s Vol.29-MPS-76 No.39 Vol.29-BIO-19 No.39 29/12/18 4 c 29 Information Processing Society of Japan

5 4 1.4π 1.2π 1.π.8π.6π.4π.2π 1.4π 1.2π 1.π.8π.6π.4π.2π.π 1.6π 1.8π 2.π.π.π.5π 1.π 1.5π 2.π type1 (w=) () () 1.6π 2.π Fig. 4 The result of numerical simulations of the phase response curve by using the spike response model with the type1 kernel, and the corresponding PRC obtained from our analysis. The solid line is the theoretical line of the phase response curve. The plus symbols indicate the numerical results. The horizontal axis shows the perturbed phase ( (, 2π]), and the vertical axis shows the phase difference. The inset is an enlargement of the region for the phase from 1.6π to 2.π. type1 type2 type1 type1 (3) κ(s) w = u(t) (14) 7mV 35mV (14) η = 55. τ η = 75. τ s = 1. w = I =.37mA ϵ =.1 (14) T T t T 1 (t ) T T 1 (t ) T (t ) (6) T (t ) 4 type1 4 () () 5.13π.1π.6π.3π.π.1π.6π.3π.π -.3π 1.6π 1.8π 2.π -.3π.π.5π 1.π 1.5π 2.π 5 type2 (w ) 4 Fig. 5 The result of numerical simulations of the phase response curve by using the spike response model with the type2 kernel, and the corresponding PRC obtained from our analysis. The notations are the same as in the Fig.4. type1 type1 type2 type2 (3) κ(s) w u(t) (14) (14) type2 τ s = 3.3 w =.2 I = 2.mA ϵ =.1 type1 type1 (14) T Vol.29-MPS-76 No.39 Vol.29-BIO-19 No.39 29/12/18 5 type2 type2 5 c 29 Information Processing Society of Japan

6 type2 1B s type2 κ(s) type2 1.6π κ(s) type1 type2 type1 type1 type2 type2 (2591(T.O.) 2521(T.A.) (M.O.)) Vol.29-MPS-76 No.39 Vol.29-BIO-19 No.39 29/12/18 1) Winfree, A.T.: The Geometry of Biological Time, Springer (199). 2) Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence, Dover (23). 3) Lengyel, M., Kwag, J., Paulsen, O. and Dayan, P.: Matching storage and recall: hippocampal spike timing-dependent plasticity and phase response curves, Nat. Neurosci., Vol.8, No.12, pp (25). 4) Preyer, A. and Butera, R.: Neuronal oscillators in aplysia californica that demonstrate weak coupling in vitro, Phys. Rev. Lett., Vol.95, No.13, p (25). 5) Ermentrout, B.: Type I membranes, phase resetting curves, and synchrony, Neural Comput., Vol.8, No.5, pp (1996). 6) Moehlis, J., Shea-Brown, E. and Rabitz, H.: Optimal inputs for phase models of spiking neurons, J. Comput. and Nonlin. Dyna., Vol.1, No.4, pp (26). 7) Ermentrout, B., Pascal, M. and Gutkin, B.: The effects of spike frequency adaptation and negative feedback on the synchronization of neural oscillators, Neural Comput., Vol.13, No.6, pp (21). 8) Stiefel, K., Gutkin, B. and Sejnowski, T.: The effects of cholinergic neuromodulation on neuronal phase-response curves of modeled cortical neurons, J. Comput. Neurosci., Vol.26, No.2, pp (29). 9) Gerstner, W.: Associative memory in a network of biological neurons, Advances in Neural Information Processing Systems 3 (Lippmann, R., Moody, J.E. and Touretzky, D.S., eds.), NIPS, Morgan Kaufmann, pp.84 9 (1991). 1) Kistler, W., Gerstner, W. and Hemmen, J.: Reduction of the Hodgkin-Huxley equations to a single-variable threshold model, Neural Comput., Vol.9, No.5, pp (1997). 11) Gerstner, W. and Kistler, W.M.: Spiking Neuron Models: Single Neurons, Populations, Plasticity, Cambridge University Press (22). 6 c 29 Information Processing Society of Japan

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