A numerical study on iso spiking bifurcations of some neural systems

A NUMERICAL STUDY ON ISO-SPIKING BIFURCATIONS OF SOME NEURAL SYSTEMS CHING MENG HUI (B.Sc.(Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF COMPUTATIONAL SCIENCE NATIONAL UNIVERSITY OF SINGAPORE 2003 Acknowledgements This thesis could not have been written without the help of a few people. I am especially grateful to the following: Prof. D. B. Creamer, for his guidance and assistance in the culmination of this thesis. My supervisors, Prof. Chow Shui-Nee, for his advice and guidance, and Dr. Deng Bo, for his help and guidance. My family for their support and encouragement. My fellow postgraduates, all the staffs and students of the Department of Computational Science, Faculty of Science, National University of Singapore. Oliver Ching i Table of Contents Acknowledgements i Table of Contents ii Summary iv Chapter Introduction 1.1 Biological Rhythms And Dynamical Systems . . . . . . . . . . . . . 1.2 Preliminaries Of Dynamical Systems And Bifurcation Theory . . . . 1.3 Preliminaries Of Neural Systems . . . . . . . . . . . . . . . . . . . . Iso-Spiking Bif. & Renormalization Uni. 10 2.1 One-Dimensional Return Map . . . . . . . . . . . . . . . . . . . . . 10 2.2 Iso-Spiking Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Scaling Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Renormalization Universality 18 . . . . . . . . . . . . . . . . . . . . . Numerical Results 25 3.1 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Numerical Results Of Model N 26 . . . . . . . . . . . . . . . . . . . . ii TABLE OF CONTENTS 3.3 3.4 iii Other Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.1 Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.2 Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.3 Model C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.4 Model D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Numerical Results of Model B . . . . . . . . . . . . . . . . . . . . . 37 Conclusion 39 Appendix A Programs A.1 Program That Runs The Model-Files . . . . . . . . . . . . . . . . . v v A.2 Model-File For Model N . . . . . . . . . . . . . . . . . . . . . . . . viii A.3 Model-File For Model A . . . . . . . . . . . . . . . . . . . . . . . . x A.4 Model-File For Model B . . . . . . . . . . . . . . . . . . . . . . . . xii A.5 Model-File For Model C . . . . . . . . . . . . . . . . . . . . . . . . xv A.6 Model-File For Model D . . . . . . . . . . . . . . . . . . . . . . . . xvii Bibliography xx Summary This thesis is based on a paper by Deng [1], and is written for readers with some background in Mathematical Analysis. We aim to show, through computer simulations, the validity of results from [1]. In Chapter One, we touch on the relationship between dynamical systems and biological science. We also introduce basic concepts and definitions in dynamical systems and neural systems. In Chapter Two, we review the paper by Deng [1]. We introduce the dynamical system that was covered in [1]. We look into the iso-spiking bifurcations of the system and, using some scaling laws and renormalization analysis, we show that the natural number is a universal constant for any model from the same family of neural systems. In Chapter Three, we present the numerical results of the simulation of the system from Chapter Two. We also detail four other models of systems from [2]. In Chapter Four, we conclude the thesis with some thoughts of the author. In the appendix, we provide programs to run simulations of the systems from Chapters Two and Three. These are all original creations by the author. iv Chapter Introduction In this chapter, we touch on the definitions of terms in dynamical systems. 1.1 Biological Rhythms And Dynamical Systems In recent years, research has shown that disorderly behaviors in biological rhythms sometimes appear to follow deterministic rules. This has led to a growing interest in using nonlinear dynamics in biology, as dynamical systems provide a way of seeing order and pattern where formerly only the random, the erratic, and the unpredictable were observed. As an example, the human body is made up of 1014 cells, especially neurons, which are believed to be the key elements in signal processing or communications. The human brain has 1011 neurons, and each has more than 104 synaptic connections with other neurons. Neurons by themselves are slow, unreliable analog units, yet working together, they can carry out highly sophisticated computations in cognition and control. By modelling these sophisticated and complex biological processes, we can study the abnormal rhythmic activity in biology systematically. These models can actu- CHAPTER 1. INTRODUCTION ally describe, to a certain level of accuracy, the actual biological systems. They exhibit spiking, bursting, chaos, and fractals, by varying parameters of the system. 1.2 Preliminaries Of Dynamical Systems And Bifurcation Theory In this section, we introduce some basic terminology of dynamical systems and bifurcation theory. Definition 1.2.1. A dynamical system is a triple {X, t, ϕ}, where X is a state space, t ∈ R and ϕt : X → X satisfies the properties ϕ0 = I, where I is the identity map on X, that is, I (x) = x for all x ∈ X, and ϕt+s = ϕt ◦ ϕs , for all t, s ∈ R. Definition 1.2.2. An orbit starting at x∗ is a subset of the state space X, Or (x∗ ) = x ∈ X : x = ϕt (x∗ ) , t ∈ R . Definition 1.2.3. A point x∗ ∈ X is called an equilibrium (fixed point) if ϕt (x∗ ) = x∗ , for all t ∈ R. Definition 1.2.4. A point x∗ ∈ X is called a periodic point if ϕt (x∗ ) = x∗ , for some t ∈ R. CHAPTER 1. INTRODUCTION f 2(x0) f (x0) x0 x0 f N -1(x0) L0 L0 Figure 1.1: Periodic orbit in a Figure 1.2: Periodic orbit in a continuous-time system. discrete-time system. Definition 1.2.5. An orbit, L0 , is a periodic orbit if each point x∗ ∈ L0 satisfies ϕt+T0 (x∗ ) = ϕt (x∗ ) , with some T0 > 0, for all t ∈ R. Figure 1.1 shows an example of a periodic orbit in a continuous-time system, while Figure 1.2 presents a periodic orbit in a discrete-time system. Definition 1.2.6. A (positively) invariant set of a dynamical system {X, t, ϕ} is a subset S ⊂ X such that x∗ ∈ S ⇒ ϕt (x∗ ) ∈ S for all t > 0. Definition 1.2.7. An invariant set S0 is stable if for any sufficiently small neighborhood U ⊃ S0 there exists a neighborhood V ⊃ S0 such that ϕt (x) ∈ U for all x ∈ V and all t > 0; Definition 1.2.8. An invariant set S0 is asymptotically stable if it is stable and there exists a neighborhood U0 ⊃ S0 such that ϕt x → S0 for all x ∈ U0 , as t → +∞. Definition 1.2.9. Given a continuous-time dynamical system x˙ = f (x) , x ∈ Rn , (1.1) where f is smooth and (1.1) has a periodic orbit L0 . Take a point x0 ∈ L0 and introduce a cross-section Σ to the orbit at this point (see Figure 1.3). An orbit CHAPTER 1. INTRODUCTION P(x) Σ x x0 L0 Figure 1.3: The Poincar´e map associated with periodic orbit L0 . starting at a point x ∈ Σ sufficiently close to x0 will return to Σ at some point x˜ ∈ Σ near x0 . Moreover, nearby orbits will also intersect Σ transversally. Thus, a map P : Σ → Σ, x → x˜ = P (x) , is constructed. The map P is called a Poincar´e map associated with the periodic orbit L0 . Definition 1.2.10 (Stable Manifold). W s (x0 ) = x : ϕt x → x0 , t → +∞ , is called the stable set of x0 . Definition 1.2.11 (Unstable Manifold). W u (x0 ) = x : ϕt x → x0 , t → −∞ , is called the unstable set of x0 . Definition 1.2.12. Given a discrete-time dynamical system x → f (x) , x ∈ Rn , (1.2) CHAPTER 1. INTRODUCTION where the map f is smooth along with its inverse f −1 . Let x0 = be a fixed point of the system and let A denote the Jacobian matrix df dx evaluated at x0 . The eigenvalues µ1 , µ2 , . . . , µn of A are called the multipliers of x0 . Mulitpliers of continuous-time dynamical systems are similarly defined. Definition 1.2.13. A fixed point is called hyperbolic if there are no multipliers on the unit circle. A hyperbolic equilibrium is called a hyperbolic saddle if there are multipliers inside and outside the unit circle. Definition 1.2.14. The appearance of a topologically nonequivalent phase portrait under variation of parameters is called a bifurcation. Definition 1.2.15. A bifurcation diagram of the dynamical system is a stratification of its parameter space induced by the topological equivalence, together with representative phase portraits for each stratum. Definition 1.2.16. The bifurcation associated with the appearance of a multiplier, µ1 = is called a fold (or tangent) bifurcation. This bifurcation is also referred to as a limit point, saddle-node bifurcation, turning point, among others. Definition 1.2.17. The bifurcation associated with the appearance of a multiplier, µ1 = −1 is called a flip (or period-doubling) bifurcation. Definition 1.2.18. The bifurcation corresponding to the presence of multipliers, µ1. = ±iω0 , ω0 > 0, is called a Hopf (or Andronov-Hopf ) bifurcation. Definition 1.2.19. The bifurcation corresponding to the presence of multipliers, µ1, = e±iθ0 , < θ0 < π, is called a Neimark-Sacker (or secondary Hopf ) bifurcation. Definition 1.2.20. An orbit Γ0 starting at a point x ∈ Rn is called homoclinic to the equilibrium point x0 of system (1.1) if ϕt x → x0 as t → ±∞. APPENDIX A. PROGRAMS tspan(2) = tspan(2) * 2; if (tspan(2) >= ttol) repeat = 0; end; else n = size(t); clear t; if (n(1) < 5) c1 = c1 + 1; y0 = evcord(end, :); if (c1 > 100) repeat = 0; end; else c2 = c2 + 1; y1 = y0; y0 = evcord(end, :); if ((norm((y1 - y0), inf) < etol) | (c2 > 50)) repeat = 0; end; clear y1; end; clear n; end; end; end; clear model epsilon rho etol ttol options tspan y0 c1 c2 repeat; vii viii APPENDIX A. PROGRAMS if (isempty(evtype) | (evtype(end) ~= 2)) evcord = []; evtype = []; end; A.2 Model-File For Model N %-=~:=~-:~-=:-~=:~=-:=-~|\+/|\+/|\+/|\+/|~-=:-=~:=~-:=-~:-~=:~=-% % Filename : neural.m % % Author : Oliver Ching % % Date : 05121998 % % Usage : % % >> [t,y]=ode15s(’neural’,[0 50],[0 .01 -.45],[],[],[]) % %-=~:=~-:~-=:-~=:~=-:=-~|/+\|/+\|/+\|/+\|~-=:-=~:=~-:=-~:-~=:~=-% function [out1, out2, out3] = neural(t, y, flag, epsilon, rho) sigma = 0.005; eta1 = 0.05; eta2 = 0.05; omega = 1.0; ccpt = 0.0; cmin = -0.5; nmax = 1.0; nmin = 0.0; vmax = 2.0; vmin = -0.5; vspk = r1 = (vmax - vspk) / (nmax - nmin); 0.0; ix APPENDIX A. PROGRAMS r2 = (nmax + nmin) / 2; r3 = (vspk - vmin) / (ccpt - cmin); if ((nargin < 4) | isempty(epsilon)) epsilon = 1.0; end; if ((nargin < 4) | isempty(rho)) rho = -0.15; end; if ((nargin < 3) | isempty(flag)) t1 = y(2) - nmin; t2 = nmax - y(2); t3 = y(3) - vmin; t4 = y(3) - rho; t5 = y(3) - vmax; t6 = y(2) - r2; t7 = y(1) - cmin; out1 = [epsilon * t4; (t1 * t2 * (t5 + r1 * t1) - eta1 * t6) / sigma; t2 * (t3 * (t3 - r3 * t7) + eta2) - omega * t1]; else switch(flag) case ’init’ out1 = [ 128]; out2 = [ 0.01 -0.45]; out3 = odeset(’RelTol’, 1e-12, ’AbsTol’, 1e-10 * ones(1, 3), ’Event’, ’ on’); case ’events’ x APPENDIX A. PROGRAMS sec1 = y(3) - (2 * vmax + vspk) / 3; sec2 = y(3) - (vspk + vmin) / 2; out1 = [sec1 out2 = [ 1]; out3 = [ -1]; sec2]; otherwise error([’Unknown flag ’’’ flag ’’’.’]); end; end; A.3 Model-File For Model A %-=~:=~-:~-=:-~=:~=-:=-~|\+/|\+/|\+/|\+/|~-=:-=~:=~-:=-~:-~=:~=-% % Filename : modela.m % % Author : Oliver Ching % % Date : 05121998 % % Usage : % % >> [t,y]=ode15s(’modela’,[0 50],[-50 0.1 0.1],[],[],[]) % %-=~:=~-:~-=:-~=:~=-:=-~|/+\|/+\|/+\|/+\|~-=:-=~:=~-:=-~:-~=:~=-% function [out1, out2, out3] = modela(t, y, flag, tauC, VCa) Cm = 1.0; gCa = 55.0; gK = 280.0; gL = 2.2; kC = 75.0; Ks = 1.0; nH = 3.0; APPENDIX A. PROGRAMS Sd = 7.5; Sn = 10.0; Ss = 10.0; tau_n = Vd = -22.0; VK = -80.0; VL = -40.0; Vn = -9.0; 0.0085; if ((nargin < 4) | isempty(tauC)) tauC = 4000.0; end; if ((nargin < 4) | isempty(VCa)) VCa = 100.0; end; if ((nargin < 3) | isempty(flag)) Vs = nH * Ss * log(y(3) / Ks); t1 = exp((Vd - y(1)) / Sd); t2 = exp((Vn - y(1)) / Sn); t3 = exp((Vs - y(1)) / Ss); dinf = / (1 + t1); ninf = / (1 + t2); sinf = / (1 + t3); taun = tau_n * t2 * ninf; t4 = (y(1) - VCa) * dinf * sinf; t5 = (y(1) - VK) * gK * y(2); t6 = (y(1) - VL) * gL; out1 = [(gCa * t4 + t5 + t6) / Cm; xi xii APPENDIX A. PROGRAMS (ninf - y(2)) / taun; -(t4 + kC * y(3)) / tauC]; else switch(flag) case ’init’ out1 = [ 128]; out2 = [ -50 0.1 0.1]; out3 = odeset(’RelTol’, 1e-12, ’AbsTol’, 1e-10 * ones(1, 3), ’Event’, ’ on’); case ’events’ sec1 = y(1) + 20; sec2 = y(1) + 50; out1 = [sec1 sec2]; out2 = [ 1]; out3 = [ -1]; otherwise error([’Unknown flag ’’’ flag ’’’.’]); end; end; A.4 Model-File For Model B %-=~:=~-:~-=:-~=:~=-:=-~|\+/|\+/|\+/|\+/|~-=:-=~:=~-:=-~:-~=:~=-% % Filename : modelb.m % % Author : Oliver Ching % % Date : 05121998 % % Usage : % xiii APPENDIX A. PROGRAMS % >> [t,y]=ode15s(’modelb’,[0 50],[-50 5.1],[],[],[]) % %-=~:=~-:~-=:-~=:~=-:=-~|/+\|/+\|/+\|/+\|~-=:-=~:=~-:=-~:-~=:~=-% function [out1, out2, out3] = modelb(t, y, flag, tauC, Vs) Cm = 1.0; gf = 60.0; gK = 110.0; gL = 25.0; gs = 25.0; kC = 2.0; Kf = 1.0; Sd = 8.0; Sm = 8.0; Sn = 8.0; tau_n = 0.026; Vd = -40.0; Vf = VK = -80.0; VL = -60.0; Vm = -18.0; Vn = -10.0; 40.0; if ((nargin < 4) | isempty(tauC)) tauC = 40.0; end; if ((nargin < 4) | isempty(Vs)) Vs = 110.0; end; if ((nargin < 3) | isempty(flag)) APPENDIX A. PROGRAMS t1 = exp((Vd - y(1)) / Sd); t2 = exp((Vm - y(1)) / Sm); t3 = exp((Vn - y(1)) / Sn); dinf = / (1 + t1); finf = Kf / (Kf + y(3)); minf = / (1 + t2); ninf = / (1 + t3); taun = ninf * t3 * tau_n; t4 = (Vf - y(1)) * gf * minf; t5 = (VK - y(1)) * gK * y(2); t6 = (VL - y(1)) * gL; t7 = (Vs - y(1)) * dinf * finf; out1 = [(t4 + gs * t7 + t5 + t6) / Cm; xiv (ninf - y(2)) / taun; (t7 - kC * y(3)) / tauC]; else switch(flag) case ’init’ out1 = [ 128]; out2 = [ -50 5.1]; out3 = odeset(’RelTol’, 1e-12, ’AbsTol’, 1e-10 * ones(1, 3), ’Event’, ’ on’); case ’events’ sec1 = y(1) + 20; sec2 = y(1) + 50; out1 = [sec1 out2 = [ sec2]; 1]; xv APPENDIX A. PROGRAMS out3 = [ -1]; otherwise error([’Unknown flag ’’’ flag ’’’.’]); end; end; A.5 Model-File For Model C %-=~:=~-:~-=:-~=:~=-:=-~|\+/|\+/|\+/|\+/|~-=:-=~:=~-:=-~:-~=:~=-% % Filename : modelc.m % % Author : Oliver Ching % % Date : 05121998 % % Usage : % % >> [t,y]=ode15s(’modelc’,[0 50],[-55 .5 0],[],[],[]) % %-=~:=~-:~-=:-~=:~=-:=-~|/+\|/+\|/+\|/+\|~-=:-=~:=~-:=-~:-~=:~=-% function [out1, out2, out3] = modelc(t, y, flag, tau_f, Vf) Cm = 1.0; gs = 200.0; gK = 250.0; gL = 13.0; Sd = 8.0; Sf = -10.0; Sn = tau_n = Vd = -18.0; VK = -80.0; VL = -60.0; 10.0; 0.0115; APPENDIX A. PROGRAMS Vn = -5.0; Vs = 40.0; if ((nargin < 4) | isempty(tau_f)) tau_f = 40.0; end; if ((nargin < 4) | isempty(Vf)) Vf = -40.0; end; if ((nargin < 3) | isempty(flag)) t1 = exp((Vd - y(1)) / Sd); t2 = exp((Vf - y(1)) / Sf); t3 = exp((Vn - y(1)) / Sn); dinf = / (1 + t1); finf = / (1 + t2); ninf = / (1 + t3); tauf = sqrt(finf) * t2 * tau_f ; taun = ninf * t3 * tau_n; t4 = (VK - y(1)) * gK * y(3); t5 = (VL - y(1)) * gL; t6 = (Vs - y(1)) * dinf * gs * y(2); out1 = [(t4 + t5 + t6) / Cm; (finf - y(2)) / tauf; (ninf - y(3)) / taun]; else switch(flag) case ’init’ out1 = [ 128]; xvi xvii APPENDIX A. PROGRAMS out2 = [ -55 0.5 0]; out3 = odeset(’RelTol’, 1e-12, ’AbsTol’, 1e-10 * ones(1, 3), ’Event’, ’ on’); case ’events’ sec1 = y(1) + 20; sec2 = y(1) + 45; out1 = [sec1 sec2]; out2 = [ 1]; out3 = [ -1]; otherwise error([’Unknown flag ’’’ flag ’’’.’]); end; end; A.6 Model-File For Model D %-=~:=~-:~-=:-~=:~=-:=-~|\+/|\+/|\+/|\+/|~-=:-=~:=~-:=-~:-~=:~=-% % Filename : modeld.m % % Author : Oliver Ching % % Date : 05121998 % % Usage : % % >> [t,y]=ode15s(’modeld’,[0 50],[-50 .5 .2],[],[],[]) % %-=~:=~-:~-=:-~=:~=-:=-~|/+\|/+\|/+\|/+\|~-=:-=~:=~-:=-~:-~=:~=-% function [out1, out2, out3] = modeld(t, y, flag, tauC, VCa) Cm = 1.0; gCa = 400.0; gKC = 9000.0; APPENDIX A. PROGRAMS gL = 25.0; kC = 2.0; Kh = 1.0; Kn = 10.0; Sd = 8.0; Sf = -10.0; Sn = 13.0; tau_f = 40.0; Vd = -13.0; Vf = -40.0; VK = -90.0; VL = -60.0; if ((nargin < 4) | isempty(tauC)) tauC = 0.06; end; if ((nargin < 4) | isempty(VCa)) VCa = 100.0; end; if ((nargin < 3) | isempty(flag)) Vn = -35 * log(y(3) / Kn); t1 = exp((Vd - y(1)) / Sd); t2 = exp((Vf - y(1)) / Sf); t3 = exp((Vn - y(1)) / Sn); dinf = / (1 + t1); finf = / (1 + t2); hinf = Kh / (Kh + y(3)); ninf = / (1 + t3); xviii APPENDIX A. PROGRAMS tauf = sqrt(finf) * t2 * tau_f; t4 = (VCa - y(1)) * dinf * hinf * y(2); t5 = (VK - y(1)) * gKC * ninf; t6 = (VL - y(1)) * gL; out1 = [(gCa * t4 + t5 + t6) / Cm; xix (finf - y(2)) / tauf; (t4 - kC * y(3)) / tauC]; else switch(flag) case ’init’ out1 = [ 128]; out2 = [ -50 0.5 0.2]; out3 = odeset(’RelTol’, 1e-12, ’AbsTol’, 1e-10 * ones(1, 3), ’Event’, ’ on’); case ’events’ sec1 = y(1) + 20; sec2 = y(1) + 50; out1 = [sec1 sec2]; out2 = [ 1]; out3 = [ -1]; otherwise error([’Unknown flag ’’’ flag ’’’.’]); end; end; Bibliography [1] Deng, B. Neural code renormalization, universal number 1, and super chaos. http://www.math.unl.edu/˜bdeng/Publications.htm, Aug. 1997 ∼ Oct. 2000. [2] Chay, T.R., Y.S. Fan, and Y.S. Lee. Bursting, spiking, chaos, fractals, and universality in biological rhythms. Int. J. Bif. & Chaos, vol. 5, no. 3, pp. 595-635, 1995. [3] Adrian, E.D. The Basis of Sensation: The Action of the Sense of Organs, W.W. Norton, New York, 1928. [4] Baesens, C., J. 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BIBLIOGRAPHY xxii [18] Poznanski, R.R. Modeling in The Neurosciences: from Ionic Channels to Neural Networks, Harwood Academic Pub., Amsterdam, The Netherlands, 1999. [19] Rieke, F., D. Warland, R. de Ruyter van Steveninck, and W. Bialek. Spikes: Exploring the Neural Code, The MIT Press, Cambridge, MA, 1996. [20] Rinzel, J. A formal classification of bursting mechanisms in excitable systems. Proc. Intern. Congr. of Mathematicians (A.M. Gleason, ed.), Amer. Math. Soc., pp. 1578-1594. [21] Sherman, A., P. Carroll, R.M. Santos, and I. Atwater. Glucose dose response of pancreatic β-cells: experimental and theoretical results. Transduction in Biological Systems, eds., C. Hidalgo, J. Bacigalupo, E. Jaimovich, and J. Vergara, pp. 123-141, Plenum Publishing Co., 1990. [22] Terman, D. Chaotic spikes arising from a model of bursting in excitable membranes. SIAM J. Appl. Math., vol. 51, pp. 1418-1450, 1991. [23] Wang, X.-J., and J. Rinzel. Oscillatory and bursting properties of neurons. The Handbook of Brain Theory and Neural Networks, M. Arbib, ed., MIT Press, pp. 686-691. [...]... Ca2+ concentration n and V are the fast variables of which n is faster for small 0 < ς 1 n corresponds to the percentage of open potassium channels and V corresponds to the cell’s membrane potential To apply an extended renormalization theory, we need to reduce the dynamics of Model N systems to a one-dimensional Poincar´ return map By the asymptotic e theory of singular perturbations, the dynamics of. .. the numerical data from the simulations, we are able to plot spike bifurcation diagrams (Figures 3.1, 3.2 and 3.3), for various values of , that support the hypothesis presented in [1] Values in Table 3.1 and Figure 3.4 (Logarithm Plot of length of iso- spiking intervals against n) also substantiate the theory discussed in Chapter Two Compiling data from the various spike bifurcation diagrams, a contour... generate an estimated numerical orbit of the system for particular set of bifurcation parametic values over a pre-defined time, t0 Then by examining this numerical orbit, we are able to determine an estimated periodic orbit for that set of bifurcation parameters Using the periodic orbit, we then obtained the spike number for that particular set of bifurcation parameters Doing this over a range of bifurcation... The linearization of R at the fixed point ψ0 has 1 as an eigenvalue and the remainder spectrum is in the open unit disk Hypothesis 2 There is an invariant codimension-one manifold W that is transversal to the weakly expanding invariant manifold U at ψ0 and the dynamics of R on W is nonexpanding Hypothesis 3 The weakly expanding manifold U intersects transversely at ψ1/2 the codimension-one manifold... Preliminaries Of Neural Systems In this section, we introduce some basic terminology associated with neural systems Definition 1.3.1 Abrupt changes in the electrical potential across a cell’s membrane are called spikes (or action potentials) Figure 1.9 shows an example of a periodic spiking system Definition 1.3.2 A neuron is quiescent if its membrane potential is at rest or it exhibits small amplitude (“subthreshold”)... only ε as our parameter and choose the decreasing direction of ε 0+ for bifurcation analysis Definition 2.2.5 Let αn be the first parametric value, such that all parametric values immediately passing it have spike number n And let ωn be the first parametric value after αn , such that there exists a burst with more than n spikes Remark This means that ωn < αn and that the system must be iso- spiking for every... generically transversal intersecting This means that irregardless of the families, the scaling laws above would hold independently, that is essentially the gist of universality This means that the scaling laws can be applied to any neural system, even those of different spike initiation and termination mechanisms For more details and proof, refer to [1] Chapter 3 Numerical Results In this chapter, we present... the numerical results of the simulation of Model N Then we discuss four other models (from [2]) for consideration 3.1 Numerical Simulations Before we present the numerical results, we will discuss how the numerical simulations were done Firstly, the mathematical model of the system that we are investigating is written as a function in a MATLAB M-file Next, using the MATLAB ODE Solver (ode15s), we can... is also equivalent to xM < c ≤ xN +1 max min Note that the system is non -iso- spiking if and only if N = M + 1 CHAPTER 2 ISO- SPIKING BIF & RENORMALIZATION UNI 16 Definition 2.2.4 Consider the return map (2.2), such that all the parameters except for ε are fixed, then we have a one-parameter family, which we will denote by fε From this point on till the end of the chapter, we will consider only ε as... Definition 2.2.6 The parameter interval (ωn , αn ] is defined as the iso- spiking interval, In Remark As ε 0+ , the number of spikes per burst increases and the iterates xn , xn+1 decreases So this means that the parameter value at which xn first max max min crosses c from above is the bifurcation point ε = αn and xn+1 passes through c min from above is the bifurcation point ε = ωn This means that we can . the iso- spiking bifurcations of the system and, using some scaling laws and renormalization analysis, we show that the natural number 1 is a universal constant for any model from the same family of. A NUMERICAL STUDY ON ISO- SPIKING BIFURCATIONS OF SOME NEURAL SYSTEMS CHING MENG HUI (B.Sc.(Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF COMPUTATIONAL SCIENCE NATIONAL. there are multipliers inside a nd outside the unit circle. Definition 1.2.14. The appearance of a topologically nonequivalent phase po rtrait under variation of parameters is called a bifurcation. Definition