Influence of Sodium Inward Current on Dynamical Behaviour of Modified Morris-Lecar Model
IInfluence of Sodium Inward Current on DynamicalBehaviour of Modified Morris-Lecar Model st Hammed Olawale Fatoyinbo
School of Fundamental SciencesMassey University
Pamerston North, New [email protected]: 0000-0002-6036-2957 nd Afeez Abidemi
Department of Mathematical SciencesFederal University of Technology
Akure, [email protected]: 0000-0003-1960-0658 rd Sishu Shankar Muni
School of Fundamental SciencesMassey University
Pamerston North, New [email protected]
Abstract —In this paper, we consider a modified Morris-Lecar model by incorporating the sodium inward current. Weinvestigate in detail the influence of sodium current conductanceand potassium current conductance on the dynamical behaviourof the modified model. Variation of sodium current conductancechanges the dynamics qualitatively. We perform a numericalbifurcation analysis of the model with sodium and potassiumcurrent conductances as bifurcation parameters. The bifurca-tion of solutions varying sodium current conductance producescomplex bifurcation structure that is not present in the existingresults of original Morris-Lecar model.
Index Terms —Excitable cells, ion conductance, Morris-Lecarmodel, period-doubling bifurcation
I. I
NTRODUCTION
The study of electrical activities in excitable cells (such asneurons, muscle cells hormones) has improved our understand-ing of electrophysiological processes in cell membranes. Thetemporal variation of cell membrane potential due to externalstimulation is known as an action potential. It arises as aresult of ion fluxes through various ion channels in the cellmembrane. The action potential is significant in physiologicalprocesses such as information transfer in neurons [1], musclecontraction [2], secretion of hormones [3]. Intensive physio-logical experiments have been carried out in investigating theunderlying mechanisms of interactions between ion channelsand action potentials.From the viewpoint of mathematics, numerous mathemati-cal models have been developed to study the nonlinear dynam-ics involved in the generation of an action potential in the cellmembrane. The most famous of them is the Hodgkin-Huxleymodel [4]. The model describes the conduction of electricalimpulses along a squid giant axon. Other well-known modelsare the FitzHugh-Nagumo model [5], [6], the Morris-Lecar(ML) model [7], and the Chay model [8].The ML model describes the electrical activities of a giantbarnacle muscle fibre membrane, despite being a model formuscle cell it has been widely used in modelling electricalactivities in other excitable cells mostly in neuron models[9] [10] [11] [12]. The two-dimensional ML model has beenextensively used in the literature despite it is an approximation of the three-dimensional ML model. Despite little attention tothe three-dimensional model, it has been used in some workrecently. Gottschalk and Haney [13] study how the activityof the ion channels are regulated by anaesthetics. The three-dimensional ML model is used by Marreiros et al [14] formodelling dynamics in neuronal populations using a statisticalapproach. Recently, Gonz´alez-Miranda [15] investigated pace-maker dynamics in the ML model using the three-dimensionalmodel. Gall and Zhou [16] considered four-dimensional MLmodel by including the second inward sodium Na + current.Based on experimental observations, the ML model is for-mulated on the assumption that the electrical activities in thecell membrane depend largely on the effect of ion fluxes viathe voltage-gated calcium Ca and voltage-gated potassiumK + ion channels in the cell membrane. In recent years,experimental and computational analyses have suggested thatsodium Na + currents are relevant in depolarisation of actionpotential in some muscle cells, for example in smooth musclecells of skeletal muscle arterioles [17] and therein references.Motivated by the results of Ulaynova and Shirokov [17],we investigate the influence of including sodium currents onthe dynamical behaviour of membrane potential using thefour-dimensional ML model introduced in [16]. In particular,we study in detail a numerical bifurcation analysis of thefour-dimensional ML model. A lot of studies on numericalbifurcation analyses have been carried out on ML model[15], [18], [19], [20], [21] however, to our knowledge apartfrom [16] there appears no work in the literature that hasconsidered the bifurcation analysis of the four-dimensionalML model. In [16], the external current is considered as thebifurcation parameter whereas in this present paper we focuson the maximal conductances of ion currents as bifurcationparameters. As a consequence, we show some additionalresults that are not present in the existing results of the MLmodel.The results of this paper are presented as follows. In Sect. II,we present the model and parameter values. The dynamics ofthe model upon variation of model parameters and detailedbifurcation analyses are carried out in Sect. III. Finally, theconclusion is presented in Sect. IV. a r X i v : . [ n li n . AO ] S e p I. M
ETHOD
The original Morris-Lecar (ML) model [7] consist of threeionic currents, Gall and Zhou modified the ML model in[16] by incorporating the inward Na + current to obtain thefollowing 4-D system: inward Na + current to obtain thefollowing 4-D system:C dVdt = I ext − I L − I Ca − I K − I Na , (1) dmdt = λ m ( V )( m ∞ ( V ) − m ) , (2) dndt = λ n ( V )( n ∞ ( V ) − n ) , (3) dwdt = λ w ( V )( w ∞ ( V ) − w ) , (4)where C is the membrane capacitance, V is the membranepotential and t is the time. The ionic currents in (1) are definedas following I L = g L ( V − v L ) , I Ca = g Ca m ( V − v Ca ) , (5) I K = g K n ( V − v K ) , I Na = g Na w ( V − v Na ) , where g L , g Ca , g K , and g Na are the maximum conductancesof the leak, calcium, potassium, and sodium channels. v L , v Ca , v K , and v Na are Nerst reversal potentials of the leak, calcium,potassium, and sodium channels. I ext is the external current.The equivalent circuit representation of the cell membranewith four ionic channels, I L , I Ca , I K , and I Na , is shown inFig. 1. g Ca g Na g K g Leak C m V Ca V Na V K V Leak
ExtracellularIntracellular V I Na I Ca I K I L Fig. 1: Equivalent circuit representation of the cell membranewith four ionic channels.The voltage dependent gating variables m , n and w corre-spond to the fraction of open calcium, potassium and sodiumchannels. The fraction of open calcium, potassium and sodium channels at steady state m ∞ , n ∞ and w ∞ , m ∞ ( V ) = 0 . (cid:18) (cid:18) V − ¯ v ¯ v (cid:19)(cid:19) n ∞ ( V ) = 0 . (cid:18) (cid:18) V − ¯ v ¯ v (cid:19)(cid:19) w ∞ ( V ) = 0 . (cid:18) (cid:18) V − ¯ v ¯ v (cid:19)(cid:19) and voltage-dependent rate constants λ m ( V ) , λ n ( V ) , and λ w ( V ) are λ m ( V ) = ψ m cosh (cid:18) V − ¯ v v (cid:19) ,λ n ( V ) = ψ n cosh (cid:18) V − ¯ v v (cid:19) ,λ w ( V ) = ψ w cosh (cid:18) V − ¯ v v (cid:19) , Unless otherwise stated, model parameters are as listed in [7]and [16]: C = 1 , I ext = 50 , g L = 2 , v L = 50 , g Ca = 4 , v Ca = 100 , g K = 8 , v K = − , g Na = 2 , v Na = 55 , v = − , v = 15 , v = 10 , v = 14 . , v = 5 , v = 3 , ψ m = 1 , ψ n = 0 . , ψ w = 0 . .III. R ESULTS AND D ISCUSSION
A. Dynamical Changes with respect to Parameter Variation
To study the dynamical behaviour of (1)–(4), we investigatethe effects of sodium (Na + ) current on the membrane potentialby varying its conductance g Na . The numerical simulations of(1)–(4) for the membrane potential V upon varying g Na areshown in Fig. 2. As shown in Fig. 2a, for low value of g Na , sin-gle action potential is observed after which the system goes toa steady state. The corresponding phase space with three statevariables: V , n and m is depicted in Fig. 2b. Upon increasing g Na , high frequency periodic oscillations of action potentialsare observed in the system (cf. Fig. 2c). The closed curvein Fig. 2d corresponds to the periodic oscillations. Furtherincreasing g Na , the system goes back to a steady state and thephase space shows that the steady state is a stable focus (cf.Figs. 2e and 2f). Similar behaviour are observed when g K and I ext are considered as varying parameters (results not shown).To gain insight on how these parameters influence systemdynamics we perform bifurcation analysis using XPPAUT, abifurcation analysis software, [22]. All figures are reproducedin MATLAB. The labels and abbreviations for the bifurcationpoints are given in Table I.TABLE I: Abbreviations and notations of bifurcation points Bifurcation AbbreviationHopf bifurcation HBSaddle-node bifurcation SNSaddle-node bifurcation of cycles SNCHomoclinic bifurcation HCperiod-doubling bifurcation PD a) (b)(c) (d)(e) (f) Fig. 2: Numerical simulations of the membrane potential V for (a) g Na = − ; (c) g Na = − ; (e) g Na = 1 . . Theircorresponding phase space are (b), (d) and (f), respectively. B. Bifurcation analysis
In this section we investigate the dynamics of model (1)–(4)through bifurcation analysiis. In Sect. III-B1 we studied theinfluence of sodium current conductance g Na on dynamics ofthe membrane potential, and the influence of potassium currentconductance is considered in Sect. III-B2.
1) Influence of g Na : Here, we investigate the influence of Na + current on action potentials through modulation of itsconductance g Na . A bifurcation diagram of the membranepotential V upon varying g Na is shown in Fig. 3. At verylow or very high values of g Na the system has a unique stableequilibrium point. As seen in Fig. 3a, the system loses stabilitythrough a subcritical Hopf bifurcation HB at g Na ≈ − . and regain stability at another subcritical Hopf bifurcation HB at g Na ≈ . . The model exhibits Type II excitabil-ity since the periodic oscillations emanate through a Hopfbifurcation [21]. Between the two Hopf bifurcation points,the system has a unique unstable equilibrium point and thereexist unstable and stable limit cycles. The unstable limit cycle generated at the first HB point gain stability through a saddle-node bifurcation of cycle SNC , and loses stability at a period-doubling PD bifurcation. At PD point, the limit cyclebifurcates into stable double-period and unstable limit cycles.The unstable limit cycle branch regains stability through asecond SNC , again loses stability through a third SNC before the limit cycle ends at HB point. The stable double-period limit cycle branch emanated at the PD loses and regainstability through further SNC bifurcations before converging tothe first unstable limit cycle branch. Continuation of the PD bifurcation results in cascade of PD bifurcations of limit cyclesas shown in Fig. 3b. The projections of the periodic trajectoriesfor Period-1, 2, 4, 8, 16 and 32 onto (V , n , m) phase space areillustrated in Fig. 4. All the double-period stable limit cyclesgenerated at each PD point undergo series of SNC bifurcationsbefore converging to the limit cycle emanated from the first HB bifurcation. The cascade of PD bifurcations of limitcycles may lead to chaotic dynamics in the system [23], [24]. (a) -20 -15 -10 -5 0 5 10 Sodium Current Conductance, g Na -150-100-50050 M e m b r a n e P o t e n ti a l , V ( m V ) SNC SNC
SNC HB HB PD
1, 2 ...32 (b) -13.4 -13.3 -13.2 -13.1 -13
Sodium Current Conductance, g Na -30-20-100102030 M e m b r a n e P o t e n ti a l , V ( m V ) HB SNC PD
1, 2 ...32
SNC SNC (c) -13.45 -13.4 -13.35 -13.3 Sodium Current Conductance, g Na -24-22-20-18-16 M e m b r a n e P o t e n ti a l , V ( m V ) SNC PD PD Fig. 3: A bifurcation diagram of the membrane potential V with g Na as bifurcation parameter. Continuous [dashed]curves correspond to stable [unstable] solutions. Black [or-ange] curves correspond to equilibria [periodic oscillations].
2) Influence of g K : Now we investigate the influence of K + current on action potentials by varying its conductance g K . Fig. 5a shows the bifurcation diagram of the membranepotential V as g K is varied. For each value of g K , there existsa unique equilibrium point. For low values and high values of g K , the equilibrium point is stable. As we increases g K , thesystem loses stability through a subcritical Hopf bifurcation HB at g K ≈ . and this led to emergence of unstablelimit cycle which becomes stable through a saddle nodebifurcation of cycles SNC at g K ≈ . . As g K increasesfurther, the stable limit cycle changes stability in another a) (b)(c) (d)(e) (f) Fig. 4: Phase-space of (1)–(4) showing the cascade of period-doubling bifurcations. (a) Period-1 (b) Period-2 (c) Period-4(d) Period-8 (e) Period-16 (f) Period-32, respectively.saddle node bifurcation of cycles
SNC at g K ≈ . become unstable, and the unstable limit cycle ends in anothersubcritical Hopf bifurcation HB at g K ≈ . . Between thetwo subcritical Hopf bifurcations, there exists a unique unsta-ble equilibrium point. Also, for . ≤ g K ≤ . and . ≤ g K ≤ . system is bistable. For these values of g K , a stable limit cycle coexist with a stable equilibrium point.Fig. 5b shows the frequency of oscillations against g K . Theonset of oscillation is at a nonzero frequency, this is typicalof oscillations that occur through a Hopf bifurcation and thistype of behaviour is classified as Type II excitability by [25].IV. C ONCLUSION
We have considered a modified ML model to explore theinfluence of second inward sodium current on the dynamics ofmembrane potential. Our results showed that upon increasingthe sodium current conductance g Na , the model move froma steady state to an oscillatory domain and further increasing g Na it returned to a steady state. Similar results was obtainedwhen g K and I ext are varied. (a)
10 20 30 40 50 60
Potassium Current Conductance, g K -50-40-30-20-10010203040 M e m b r a n e P o t e n ti a l , V ( m V ) SNC HB SNC HB (b) Potassium Current Conductance, g K F r e qu e n c y ( H z ) SNC SNC Fig. 5: (a) A bifurcation diagram of the membrane potential V with g K as a bifurcation parameter (b) A plot of frequencyas a function g K . Continuous [dashed] curves correspond tostable [unstable] solutions. Black [orange] curves correspondto equilibria [periodic oscillations]The motivation of this work was to investigate the effectof conductances of ion currents on change in the membranepotential of the modified ML model. We showed how variationof sodium current conductance g Na and potassium currentconductance g K affect the dynamics of the membrane potentialvia bifurcation analysis.We revealed the modified ML model exhibits Type IIexcitability as we varied either g Na or g K , We found that uponvariation of sodium current conductance g Na , the modifiedmodel exhibits period-doubling PD bifurcation which is notpresent in existing results of the two-dimensional and three-dimensional ML model. The existence of PD bifurcations isan indicator that the modified ML model can exhibit chaoticbehaviour in some parameter regime.The results in this paper may explain further the experi-mental results of the impacts of sodium current on membranedepolarisation process. Moreso, our results could be useful inunderstanding diseases associated with ion channel conduc-tance and in therapeutics.Although Gall and Zhou in [16] considered the bifurcationanalysis of the modified ML model with I ext as a bifurcationparameter, their bifurcation diagram seems incomplete, thusin future work, we will consider the influence I ext on thedynamical behaviour of the modified ML model, and givea detailed bifurcation structure as I ext varies. Also, we willconsider two-parameter bifurcation analysis to explore thecomplex behaviours observed in the modified ML model.R EFERENCES[1] A. Mondal, R. K. Upadhyay, J. Ma, B. K. Yadav, and S. K. Sharma,“Bifurcation analysis and diverse firing activities of a modified excitableneuron model,”
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