Chaotic Bursting in Single Neurons Jon Newman¤ Department of Bioengineering Georgia Institute of Technology, Atlanta, GA 30332-0430, U.S.A (Dated: December 7, 2008) It has become fashionable to report on the existence of deterministic chaos in biophysical neuron models. Much of the time, the importance of this behavior is given as self-evident. Moreover, it is rarely transparent whether these studies actually investigate a chaotic behavior since a full mathematical analysis is not pursued. This report revisits a biophysical paper reporting chaotic bursting in an intrinsically bursting neuron model ref. [1]. Using cycle-expansion techniques, we determine whether these models in fact produce chaotic oscillations and how these oscillations come to be. We related these results to biological mechanisms that may be responsible for the chaotic dynamics in neurons. Finally, we investigate how likely chaotic bursting is to play a functional role in single neurons given the presence of large noise in the vast majority of biological systems. PACS numbers: 87.19.ll, 05.45.Ac, 05.10.{a Keywords: neuronal bursting, chaos, cycle-expansions 0. DISCLAIMER II. STATE SPACE ANALYSIS OF BURSTING Dynamical bursting models are a system of ordinary di®erential equations of the form This is a term report for the class `Chaos and What To Do About It' taught by Predrag Cvitanovi¶cat the Georgia x_ = f(x; y); (1) Institute of Technology during the Fall semester of 2008. y_ = ²g(x; y); x 2 Rm; y 2 Rn; (2) The content has not passed any rigorous peer review and may contain errors. where 0 · ² is a small parameter. (1{2) are singularly perturbed (SP) di®erential equations. Thus, using ge- ometric singular perturbation methods ref. [1], the dy- namics of bursting models can be explored by decom- posing the full system into fast- and slow-subsystems: (1) and (2), respectively. Here, the slow-subsystem can act independently, be a®ected synaptically, or interact I. INTRODUCTION locally with the spiking fast-subsystem to produce alter- nating periods of spiking and silence in time. To examine There is little doubt that chaotic oscillations abound the dynamical mechanism implicit to a certain bursting in neural systems. After all, even the smallest nervous behavior, y is treated as bifurcation parameters of the systems are composed of thousands of coupled oscilla- fast-subsystem. This is formally correct in the singular tors that are highly non-linear, but obviously bounded system for ² = 0, but is reasonable when there is large to some functional regime. Many studies, however, claim time separation between fast and slow dynamics. One that functional chaos arises in single neurons that exhibit should note that when ² 6= 0, the existence of topological `bursting' behavior ref. [1, 2]. Here we examine why this objects, such as equilibria and limit cycles, and events, claim is true, and why it is probably meaningless. such as bifurcations, of the isolated fast subsystem do not actually exist; they are merely estimates of system Bursting is a dynamic state characterized by alternat- dynamics that converge to true dynamics as ² ! 0. ing periods of activity and quiescence. Many dynamical An bene¯t of this fast-slow perturbation analysis is systems demonstrate bursting behavior. For instance, that it allows an objective categorization of bursting nerve cells can exhibit autonomous or induced bursting mechanism by the type of pseudo-bifurcation to and from by ¯ring separated groups of action potentials in time. the active phase. The Chay-Keizer model ref. [6] was the Autonomous bursting neurons are found in a variety ¯rst single neuron model to report on intrinsic aperiodic of neural systems, from the mammalian cortex ref. [3] spiking and bursting behavior ref. [1]. Using the topologi- and brainstem ref. [4] to identi¯ed invertebrate neurons cal categorization scheme, this model is a fold/homoclinic ref. [5]. burster - it enters the spiking regime via a fold of equi- libria and leaves when the stable periodic orbit describ- ing spiking is annihilated via collision with an unstable equilibrium. The Chay-Keizer model is a ¯ve param- ¤Electronic address: [email protected] eter model with four variables dictating fast dynamics 2 m = 0.04 m = 0.038 m = 0.03697 10 10 10 (a) (b) (c) 10 8 8 8 v 6 6 6 c 4 4 4 2 2 2 8 0 0 0 -2 -2 -2 1.0 1.5 2.0 1.0 1.5 2.0 1.0 1.5 2.0 m = 0.0369 m = 0.03688 m = 0.0362 6 10 10 10 (d) (e) (f) 8 8 8 6 6 6 v 4 4 4 4 2 2 2 0 0 0 -2 -2 -2 2 1.0 1.5 2.0 1.0 1.5 2.0 1.0 1.5 2.0 v m = 0.03613 m = 0.03 m = 0.02 r 10 10 10 (g) (i) 8 8 (h) 8 0 6 6 6 4 4 4 2 2 2 0 0 0 -2 -2 -2 -2 1.0 1.5 2.0 1.0 1.5 2.0 1.0 1.5 2.0 1.0 1.5 2.0 u FIG. 2: Period doubling route to a chaotic regime in ¹, as wit- nessed in the Chay-Keizer model, is con¯rmed in the canoni- FIG. 1: Solution to (2) and (3) with parameters I = 1; ¹ = cal model.(a)-(e) period doubling route through 1-, 2-, 4-, 8-, 0:02; vc = 10; vr = 1; d = 0:2. With the exception of ¹, 16-periodic solutions. (f) Aperiodic spiking. (g) Aperiodic these parameters are held constant in this report. v escapes bursting. (h,i) Stable bursting. from silence via fold of equilibria shown as a green dot. The stable equilibrium, vsq (bold line) de¯nes the resting potential during silence. After the fold, periodic spiking is enforced by we use a introduced by Lorenz in the examination of his resets, shown by lines with arrows, until vr = vuq and v falls back to vsq via a homoclinic orbit bifurcation, shown as a red attractor by recording spike to spike values of u. For ¹ = dot. 0:0362, this results in the mildly discouraging unimodal map shown in Fig. 3(a). We will refer to this map as the U(u). and a single slow variable. This system can be reduced Iterated unimodal maps, like U, are well under- to a canonical form while preserving the fold/homoclinic stood. The non-wandering set of U resides in M ¼ bursting mechanism. State equations are [1:67204; 1:83674]. M can be partitioned into two seg- 2 v_ = I + v ¡ u; (3) ments M0 ¼ [1:67204; 1:8), M1 ¼ (1:8; 1:83674] and a u_ = ¡¹u; (4) symbolic dynamics de¯ned by ½ where I is a constant current and ¹ is a small positive 0 if ui 2 M0 si = (6) constant. The system is reset after a voltage spike by, 1 if ui 2 M1: if v = vc; (v à vr; u à u + d); (5) + Using (6), an in¯nite symbol sequence, S (u0) = where vc ¡ vr and d are discrete shifts in variables that s1s2s3::: represents a unique future itinerary for each account for hyperpolarization and voltage coupling of u, u0 2 M. Kneading theory ref. [7] provides a simple way respectively. Using the slow variable upas a bifurcation to relate U to a topologically-conjugate piecewise linear parameter of (3), the equilibria of v are u ¡ I. A topo- tent-map on the unit interval. A kneading sequence for U + ¤ ¤ logically normal saddle-node bifurcation occurs at u = I. is K = S (u0) where u0 = 1:80 is the critical point that When u < I, v ! vc like tan(t) and a reset occurs. A provides the maximum image under U. K is converted b typical bursting solution is plotted in Fig. 1. to the kneading point, kb = 0:w1w2w3::: by ½ wi if si+1 = 0 III. ENTER CHAOS? wi+1 = (7) 1 ¡ wi if si+1 = 1; The Chay-Keizer model displays a regime aperiodic where w1 = s1. Finally, kb is converted to the kneading spiking in a parameter representing a time constant of value k via a binary expansion. This value is then used the slow variable, equivalent to ¹ in (3). As the param- to create the dike-map, eter corresponding to ¹ was decreased in their model, 8 a transition from periodic spiking to bursting occurred < 2x if x 2 [0; k=2) via a period doubling sequence to a chaotic regime. We f(x) = k if x 2 [k=2; 1 ¡ k=2] (8) encounter the same behavior in our simpli¯ed model, as : ¡2(1 ¡ x) if x 2 (1 ¡ k=2; 1]: shown in Fig. 2. If one compares U with (7), it becomes apparent that if f(S+) > f(K) the the future itinerary S+ is inadmissi- IV. SYMBOLIC DYNAMICS + ble in U since the image of the u0 corresponding to S ¤ exceeds u0. With ¹ = 0:0362, we estimated k ¼ 0:9147::: It is possible (and convenient) to further reduce (3{ for a ¯nite K of 50 or so kneading segments to create the 4) to a one dimensional discrete time map. To do this, dike-map shown in Fig. 3. 3 ℳ ℳ ℳ ℳ 0 1 1.0 0 1 (a) ku TABLE I: Admissible prime cycles of U. 1.80 Length Cycle Length Cycle Length Cycle 1 1 + + 1 1 2 01 3 001 n 1.75 n u x 011 4 0111 5 00101 6 001011 1.70 00111 010111 01011 011111 1.70 1.75 1.80 0 1.0 01111 u x n n 7 0010101 8 00100101 9 001001011 0010111 00100111 001010101 FIG.