"In this contribution two results are taken: (1) The synchronization of noiseless Hodgkin- Huxley (HH) neurons is possible from robust feedback based on Lie algebra approaches and (2) the fact that, from Lie algebra of vector fields, the generalized synchronization of different (triangular form) chaotic systems can be used to derive an explicit synchronization function. Both results are extended to derive the synchronization function in HH neurons despite this systems are not in triangular form. Thus, the Lie algebra of vectors fields permits to establish a theoretical framework for finding the synchroniza- tion function in chaotic systems in face they have different model.

Barajas Ramírez, Juan Gonzalo

Femat Flores, Alejandro Ricardo

Solís Perales, Gualberto Celestino

English

Repositorio IPICYT

Este artículo puede ser usado únicamente para uso personal o académico. Cualquier otro uso requiere permiso del autor. Lie algebra on synchronization of different systems: a generalized functionfor Hodgkin-Huxley neuronsEXTENDED ABSTRACTR. Femat∗, J.G. Barajas-Ramı́rez†and G. Soĺıs-Perales‡Laboratorio para Biodinámica y Sistemas AlinealesMatemáticas Aplicadas, IPICyTApdo. Postal 3-90, Tangamanga 78231San Luis Potośı, S.L.P., MéxicoSummary: In this contribution two results are taken: (1) The synchronization of noiseless Hodgkin-Huxley (HH) neurons is possible from robust feedback based on Lie algebra approaches and (2) thefact that, from Lie algebra of vector fields, the generalized synchronization of different (triangular form)chaotic systems can be used to derive an explicit synchronization function. Both results are extended toderive the synchronization function in HH neurons despite this systems are not in triangular form. Thus,the Lie algebra of vectors fields permits to establish a theoretical framework for finding the synchroniza-tion function in chaotic systems in face they have different model.Motivation: For several decades many attempts have been addressed to understand the processingof biological information in single neurons and neural networks. Experimental reports [1] suggest thatthe synchronization plays a very important role in the processing of information by large ensembles ofneurons. Recently, it has been demonstrated that a minimal ensemble of two coupled living neurons firesynchronized spiking activity when depolarized by an external DC current [2]. However, total neuralmechanisms underlying synchronization are not well understood yet. The Hodgkin-Huxley (HH) neuronsare usually used as realistic models of neuronal systems, for studying neuronal synchronization. Sometheoretical approaches investigate the synchronization phenomena considering diffusive coupling and theinfluence of intrinsic noise as a promoter of neuronal activity [3], and studying the synchronizationdynamics related to the rhythmic oscillations phenomena (theta and gamma frequency rhythms) inneurons of localized areas of the brain [4]. In addition, the forcing of HH neurons by external stimulushas been widely studied [5] for tonic or periodic currents that trigger the action potential displaying spikeactivity and refractory dynamics.The efforts have been focused on the analysis of the time-invariant manifolds related with synchro-nization [9]-[11]. Two basic approaches have been exploited. On the one hand, chaotic synchronizationhas been interpreted as the prediction of the chaotic system, i.e., observability approach. In this sense,the reconstruction of the drive system attractor from the response system is interpreted as an observer[11]. An interesting point about observability of the synchronization systems is that differential geometryallows to find an invariant space under vector fields where the attractor can be reconstructed. Syn-chronization of chaotic systems, on the other hand, has been also studied from the measurable variables(system output) [8]-[10]. In such a case, the synchronization is understood as a stabilization problem.In other words, to compute the control input such that the difference between trajectories of the slavesystem xS(t) remains close to the trajectories of the master system xM (t). That is, the point is to findthe invariant space such that the origin of the synchronization error system ‖xM (t) − xS(t)‖ = 0 canbe stabilized. Both observability and controlability of nonlinear systems are included in the geometricalcontrol theory [13].∗Corresponding author. rfemat@ipicyt.edu.mx†jgbarajas@ipicyt.edu.mx‡División de Computación y Electrónica, CUCEI, Universidad de Guadalajara. Av. Revolución 1500, Guadala-jara, Jal., México. gualberto.solis@cucei.udg.mx1Here, the goal is to provide some remarks on the diffeomorphism between chaotic attractors by ex-ploiting the Lie algebra of the chaotic systems [12] to compute the synchronization function. In thiscontribution, two previous results are taken to study the synchronization between HH neurons: the Lie-based approach to the generalized synchronization of different systems [14] and the fact that two HHneurons can be synchronized via robust asymptotic feedback [15] That is, we are interested in the Lie-based geometric properties of the class of dynamical system given by ẋ = τM (xM ) − τS(xS) − g(x)u,where x ∈ Rn is, by definition, x := xM − xS , and stands for the state vector of the synchronizationerror system; τM : Rn → Rn, τS : Rn → Rn and g : Rn → Rn are smooth vector fields. The productg(x)u is related to the synchronization command. The scalar function u = u(x) can be computed fromthe construction of accessibility spaces, and is named the control input [13]. Thus, we are interested infinding the output y = λ(x) , λ(xM − xS) of the above-mentioned dynamical system such that synchro-nization is attained. Thus the synchronization problem is formulated in terms of geometrical propertiesof the system along the vector fields τ(x, xM ) = τM (xM )− τS(xS) and g(x) as: what are the geometricalproperties of any synchronization error system such that the scalar function u = u(x) guarantees theexistence of the synchronization (output) function y = λ(x)? Such problem is addressed via Lie algebraof the vector field related to the synchronization error system.Main results: The following set of four coupled nonlinear differential equations represents the HHneuron model [6]:CmdVdt= Iext − gKn4 (V − VK) − gNam3h (V − VNa) − gl (V − VNa) , (1)dPKdt= αPK (V ) (1 − PK) − βPK (V )PK , (2)dPNadt= αPNa (V ) (1 − PNa) − βPNa (V )PNa, (3)dINadt= αINa (V ) (1 − PNa) − βh (V ) INa, (4)where variables V , PK , PNa and INa represent the membrane potential, the activation of the potassiumflow current, the activation and inactivation of the sodium flow current, respectively. Cm is the membranecapacitance, gK , gNa and gl are the maximum ionic and leak conductances, while VK , VNa and Vl standfor the ionic and leak reversal potentials. The external stimulus current can be modeled by the term Iext,usually a tonic or periodic forcing. The explicit form of the functions αj(V ) and βj(V ) (j = n, m, h) inEqs. (2)-(4), and nominal values for the system parameters can be found in [6, 7]. Our starting point isbased on the following facts [13]:• Fact 1 : Consider an affine nonlinear system ẋ = f(x) + g(x)u; where x ∈ Ω ⊆ Rn, u ∈ R,g, f : Rn → Rn are smooth vector fields. Besides, let us consider that y = h(x) for any smoothfunction h(x). If involutivity condition is satisfied, then the mappings Φ1 : Rn → Rρ, x 7→ zand Φ2 : Rn → Rn−ρ, x 7→ (z, ν) are such that the affine nonlinear system can be written in thecanonical formżi = zi+1, i = 1, 2, ..., ρ− 1,żρ = α(z, ν) + β(z, ν)u, (5)ν̇ = ζ(z, ν),and can be derived from Lie derivatives of the output function h(x) along the vector fields f(x)and g(x) as followsz = Φ1(x) =h(x)Lfh(x)...Lρ−1f h(x)(6)2andν = Φ2(x) =φρ+1(x)φρ+2(x)...φn(x), (7)moreover, it is always possible to chose φρ+1, ..., φn in such a way thatLgφj(x) = 0, ρ + 1 ≤ j ≤ n. (8)The Fact 1 is well known in nonlinear control theory. Here it is resumed for clarity in presentationand exploited in neuronal synchronization towards robust feedback synchronization of HH neurons.• Fact 2 : If exists the map Φ = (Φ1, Φ2) : Rn → Rn, x 7→ (z, ν) derived from (6) and (7), then thereexists the inverse Φ−1(Φ(x)) = x ∈ Ω ⊂ Rn. This fact is proved since h(x), Lfh(x), ..., Lρ−1f h(x)and φρ+1(x), ..., φn(x) are linearly independent at any x in the neighborhood U ⊂ Ω ⊆ Rn of thepoint x0 in Ω.Firstly, let us define the state variables by x = (V, PK , PNa, INa) ∈ Ω ⊂ R4. Then, by assuming theoutput function is an scalar given by the membrane potential (i.e., h(x) = x1), the dynamical model fora HH neuron can be written in nonlinear affine form ẋ = f(x) + g(x)u, withf(x) ={1/Cm[Iext − gKx42 (x1 − VK)−gNax33x4 (x1 − VNa) − gl (x1 − VNa)]}αn(x1)(1 − x2) − βn(x1)x2αm(x1)(1 − x3) − βm(x1)x3αh(x1)(1 − x4) − βh(x1)x4, g(x) =1000, (9)and the output y = h(x) is given by the membrane potential, i. e., h(x) = x1. By computing the Liederivatives of the output function along the vector fields (9), we obtainz = Φ1(x) = h(x) = x1 (10)andν = Φ2(x) =φ1(x)φ2(x)φ3(x) , (11)where φ1, φ2 and φ3 are solutions of the differential equations given by (8); that is, for this specific case,we have∂φj∂x1+∂φj∂x2+∂φj∂x3= 0 (12)where j = 1, 2, 3. If the PDE’s (12) have solution there exists a diffeomorphic function Φ = (Φ1, Φ2)T :Ω → R4, x 7→ (z, ν), such that the dynamics of the synchronization error can be written in the form(5). As matter of fact, a solution for (12) is found for any (φ1, φ2, φ3) 6= (φ1(x1), φ2(x1), φ3(x1)); hencethe function Φ exists. Finally, if there exists a synchronization force u such that the system of thesynchronization error is asymptotically stable, the synchronization function becomes xs = Ψ , Φ−1S ◦ΦM (xM ), where subscripts M and S denote the mater and slave neurons, respectively [14]. It shouldbe noted that, generally speaking, the solution of (5) is not unique. Hence the synchronization functionΨ : ΩS → ΩS , xM 7→ xS is a smooth vector field whose components φ1, φ2φ3 are exist but can beuncertain.In order to show this fact, we consider the following solution: φ(j,k) = xj,k, where j = 2, 3, 4 andk = M, S. Assuming the output is the membrane potential; i.e., h(x) = x1. Then, the model of theMaster and slacve HH neurons can be transformed intoż1 = 1/Cm[Iext − gKν41 (z1 − VK) − gNaν32ν4 (z1 − VNa)−gl (z1 − VNa)] + u , (13)ν̇1 = αn (z1) (1 − ν1) − βn (z1) ν1 , (14)ν̇2 = αm (z1) (1 − ν2) − βm (z1) ν2 , (15)ν̇3 = αh (z1) (1 − ν3) − βh (z1) ν3 . (16)3Then, both master and slave neurons are separately transformed and the maps ΦM (xM ) and ΦS(xS) arederived to get(zMνM)=(Φ1M (xM )Φ2M (xM ))and(zSνS)=(Φ1S(xS)Φ2S(xS)), (17)from where each HH neuron can be transformed into (5) which implies that there is a driving sig-nal synchronizing the master behavior onto slave neuron [15]. Then, if stability holds and neuronsare minimum-phase systems, (zS , νS) → (zM , ν∗S) for t > t0 ≥ 0 and initial conditions (z(0), ν(0)) =(Φ1(x(0)), Φ2(x(0))) in physical domain. Note that ν∗S is a stable manifold which can correspond to thestable manifold of the master neuron ν∗M . In this case complete synchronization is achieved. In caseν∗S 6= ν∗M , the partial state synchronization is attained [16].Preliminary conclusions and discussion: The composition Φ−1S (Φ1(xM ); ν∗S) = xS ∈ Ω ⊂ Rn,where Ω denotes the physical domain. In particular, if ν∗S ≡ ν∗M for all time t > t0 ≥ 0, where t0 standsfor time of turning on the control, then xS = Φ−1S (Φ1M (xM ), Φ2M (xM )). Since HH neurons (1)-(4) areminimum-phase systems, the GS yields the following function xS =(h−1S (hM (xM )), x∗2S , x∗3S , x∗4S)T . Itshould be noted that the explicit expression for the synchronization function Ψ depends on the solutionof (12). Then, the partial state synchronization is depicted but a question arises: how can we knowabout the geometric or algebraic properties of the synchronization function depends on multiple solutionof (12)? This question is under study and it will be reported elsewhere.References[1] A. K. Kreiter, W. Singer, J. Neurosci., 16 (1996) 238196.[2] R. C. Elson, A. I. Selverston, R. Huerta, N. F. Rulkov, M. I. Rabinovich, H. D. Abarbanel, Phys.Rev. Lett., 87 (1999) 5692.[3] J. M. Casado, J. P. Baltanás, Phys. Rev. E, 68 (2003) 061917.[4] G. B. Ermentrout, N. Kopell, Proc. Natl. Acad. Sci., 95 (1998) 1259.[5] Y. Yu, W. Wang, J. Wang, F. Liu, Phys. Rev. E, 63 (2001) 21907.[6] A. L. Hodgkin, A. F. Huxley, J. Physiol. (London) 117 (1952) 500.[7] C. Koch, Biophysics of Computation: Information Processing in Single Neurons (Oxford UniversityPress, New York, 1998).[8] R. Femat and J. Alvarez-Ramirez, Phys. Letts. A, 236, (1997), 307.[9] S. Boccaletti, L.M Pecora and A. Pelaez, Phys. Rev. E, 63 (2001) 066219.[10] K. Josić, Nonlinearity,13,(2000) 1321.[11] H. Nijmeijer, and M. Y. Mareels, IEEE Trans. Circuits and Syst. I., 44 (1997) 882.[12] L. Kocarev, U. Parlitz, and B. Hu, Chaos, Solitons and Fractals, 9 (1998) 1359.[13] H. Nijmeijer and A. van der Schaft, Nonlinear control dynamical systems, Springer-Verlag, 1990.[14] R. Femat, L. Kocarev, L. van Gerven, M.E. Monsivais-Pérez, Phys. Letts A, 342 (2005) 247.[15] O. Cornejo-Pérez, R. Femat, Chaos Solitons and Fractals, 25 (2005) 43.[16] R. Femat, G. Soĺıs-Perales, Phys. Letts A, 262 (1999) 50.4

Lie algebra on synchronization of different systems: a generalized function for Hodgkin-Huxley neurons

https://repositorio.ipicyt.edu.mx///bitstream/11627/4020/1/Physcon07.pdf

Lie algebra on synchronization of different systems: a generalized function for Hodgkin-Huxley neurons

Abstract

Similar works

Full text

Available Versions

Repositorio IPICYT