Computational Neuroscience. Session 4-3roquesol/Computational_Neuroscience_Summer... · Session 4-3 Cable theory II . The Morris-Lecar Model. Stability of Fixed Points Dynamical Systems

Dynamical Systems

Computational Neuroscience. Session 4-3

Dr. Marco A Roque Sol

06/25/2018

Dr. Marco A Roque Sol Computational Neuroscience. Session 4-3

Dynamical Systems Two-Dimensional Systems.

Cable theory II .

Dynamical Systems.

The Morris-Lecar Model

One of the simplest models for the production of actionpotentials is a model proposed by Kathleen Morris andHarold Lecar . The model has three channels: apotassium channel, a leak, and a calcium channel.

In the simplest version of the model, the calcium currentdepends instantaneously on the voltage. Thus, theMorris-Lecar equations have the form



Cable theory II .

Dynamical Systems.


cMdVMdt = −gK n4(V − EK )− gL(V − EL)− gCa(L)m∞(V − ECa)

= Iapp − Iion(V ,n)

dndt = φ (n∞(V )− n) /τn(V )



Cable theory II .

Dynamical Systems.

The Morris-Lecar Modelwhere

m∞(V ) = 12 [1 + tanh([V − V1]/V2)]

τn(V ) = 1cosh([V−V3]/(2V4))

n∞(V ) = 12 [1 + tanh([V − V3]/V4)]

Here, V1,V2,V3, and V4 are parameters chosen to fitvoltage-clamp data.



Cable theory II .

Dynamical Systems.

The solutions shown in the figure below demonstrate thatthe Morris-Lecar model exhibits many of the propertiesdisplayed by neurons.


Cable theory II .

Dynamical Systems.

Cable theory II .Dynamical Systems.

Here, the parameters are listed in Table below theHopf case.


Cable theory II .

Dynamical Systems.


The figure above-left demonstrates that the model isexcitable if Iapp = 60. That is, there is a stable constantsolution corresponding to the resting state of the modelneuron.

A small perturbation decays to the resting state, whereas alarger perturbation, above some threshold, generates anaction potential.The solution (V1(t),n1(t)) = (VR,nR) isconstant; VR is the resting state of the model neuron.



Cable theory II .

Dynamical Systems.


The solution (V2(t),n2(t)) corresponds to a subthresholdresponse. Here, V2(t) = 0 is slightly larger than VR and(V2(t),n2(t)) decays back to rest. Finally, (V3(t),n3(t))corresponds to an action potential. Here, we start withV3 = 0 above some threshold. There is then a largeincrease of (V3(t) followed by (V3(t) falling below VR andthen a return to rest.



Cable theory II .

Dynamical Systems.


The figure bellow-right shows a periodic solution of theMorris-Lecar equations. The parameter values are exactlythe same as before; however, we have increased theparameter Iapp, corresponding to the applied current. If weincrease I app further, then the frequency of oscillationsincreases; if Iapp is too large, then the solution approachesa constant value.


Cable theory II .

Dynamical Systems.



Cable theory II .

Dynamical Systems.

The Morris-Lecar Model. The Phase Plane

In the following, we will show how dynamical systemsmethods can be used to mathematically analyze thesesolutions.

it will be convenient to write the previous equations as

dVdt = f (V ,n)

dndt = g(V ,n)



Cable theory II .

Dynamical Systems.


The phase space for this system is simply the (V ,n) plane;this is usually referred to as the phase plane. If .V .t/; n.t//is a solution of (3.2), then at each time t 0 , (V ,n)(V (t0),n(t0))defines a point in the phase plane. The pointchanges with time, so the entire solution (V (t),n(t)) out acurve (or trajectory or orbit) in the phase plane.



Cable theory II .

Dynamical Systems.


What is special about solution curves is that the velocityvector at each point along the curve is given by theright-hand side of above set of equations.That is, thevelocity vector of the solution curve (V (t),n(t)) at a point(V0,n0) is given by (V ′(t),n′(t)) = (f (V0,n0),g(V0,n0)).

This geometric property - that the (f (V ,n),g(Vn)) alwayspoints in the direction that the solution is flowing -completely characterizes the solution curves.



Cable theory II .

Dynamical Systems.


Two important types of trajectories are fixed points(sometimes called equilibria or rest points) and closedorbits. At a fixed point, f (VR,nR) = g(VR,nR) = 0 thiscorresponds to a constant solution. Closed orbitscorrespond to periodic solutions. That is, if (ν(t), η(t))represents a closed orbit, then there exists T > 0 such that(ν(t + T ), η(t + T )) = (ν(t), η(t)) for all t .

A useful way to understand how trajectories behave in thephase plane is to consider the nullclines. TheV − nullcline is the curve defined by V ′ = f (V ,n) = 0 andthe n− nullcline is where n′ = g(V ,n) = 0.



Cable theory II .

Dynamical Systems.

The Morris-Lecar Model. Stability of Fixed Points

Note that along the V − nullcline, the vector field(f (V ,n),g(V ,n)) points either up or down, and along then− nullcline, vectors point either to the left or to the right.Fixed points are where the two null-clines intersect.

One can determine the stability of a fixed point byconsidering the linearization of the vector field at the fixedpoint. The linearization of the diferential equations at afixed point (VR,nR) is the matrix



Cable theory II .

Dynamical Systems.


M =

∂f∂V (VR,nR)

∂f∂n (VR,nR)

∂g∂V (VR,nR)

∂g∂n (VR,nR)



Cable theory II .

Dynamical Systems.


The fixed point is stable if both of the eigenvalues (vectors x such that satisfy the equation Mx = λx ; x iscalled the eigenvector and λ is the eigenvalue of M ) of thismatrix have a negative real part; the fixed point is unstableif at least one of the eigenvalues has a positive real part.For the Morris-Lecar equations, the linearization is given by

M =

∂Iion(VR ,nR)∂V /CM −gK (VR − EK )/CM

φn′∞(VR)/τn(VR) −φ/τn(VR)

=

a b

c d



Cable theory II .

Dynamical Systems.


in fact,

a =1

CM

∂

∂VIion(VR,nR) = (−gL − gK nR − gCam∞(VR))

+(ECa − VR)gCam′∞(VR)/CM

We now find conditions on the nonlinear functions in theprevious equation for when the fixed point is stable.



Cable theory II .

Dynamical Systems.


Suppose the equilibrium voltage lies between EK and ECa,a reasonable assumption. Then b < 0, c > 0, and d < 0 inthe linearization. Only a can be either negative or positiveand the only term contributing to the positivity of a is theslope of the calcium activation function, m∞(VR.



Cable theory II .

Dynamical Systems.


If a < 0, then the fixed point is asymptotically stable sincethe trace of M is negative and the determinant is positive.(Recall that the trace is the sum of the eigenvalues and thedeterminant is the product of the eigenvalues.) Note thatthe slope of the V − nullcline near the fixed point is givenby −a/b. Since b < 0, it follows that if this slope isnegative, then the fixed point is stable; that is, if the fixedpoint lies along the left branch of the V − nullcline, then itis stable



Cable theory II .

Dynamical Systems.


Now suppose the fixed point lies along the middle branchof the V − nullcline, so a > 0. Note that the slope of then− nullcline, −c/d , is always positive. If the slope of theV − nullcline is greater than the slope of the n− nullcline(i.e., −a/b > −c/d ), then ad − bc < 0.In this case, thedeterminant is negative and the fixed point is an unstablesaddle point.



Cable theory II .

Dynamical Systems.


In contrast, if the slope of the n-nullcline is greater thanthat of the V − nullcline, then the fixed point is a node or aspiral. In this case, the stability of the fixed point isdetermined by the trace of M: the fixed point is stable ifa + d < 0 and it is unstable if a + d > 0.

Since a > 0 and d = −φ/τn(VR), it follows that the fixedpoint is unstable if φ is sufficiently small. Note that φ

governs the speed of the potassium dynamics.

Recall that for the parameters given in the Table below


Cable theory II .



Cable theory II .

Dynamical Systems.

The Morris-Lecar Model. Excitable Systems

for the Hopf case, the system is excitable if Iapp = 60. As in thefigure below-a demonstrates, a small perturbation in voltagefrom the resting state decays back to rest, whereas asufficiently lage perturbation in voltage continues to increaseand generates an action potential.


Cable theory II .

Dynamical Systems.Excitable Systems


Cable theory II .

Dynamical Systems.


Phase plane analysis is very useful for understanding whatseparates the firing of an action potential from thesubthreshold return to rest in this model. The projection ofthe solutions shown in below-a


Cable theory II

Dynamical Systems. Excitable Systems


Cable theory II .

Dynamical Systems.


Note that the V − nullcline separates points alongtrajectories in which V < 0 and V > 0. In particular, Vincreases below the V − nullcline and V decreases abovethe V − nullcline. We further note that the V − nullcline is“cubic”.

This suggests a perturbation from rest that lies to the “left”of the middle branch of the V − nullcline will return quicklyto rest, whereas a perturbation that lies to the “right” of themiddle branch of the V − nullcline will initially display anincrease in membrane potential, corresponding to anaction potential, before returning to rest.



Cable theory II .

Dynamical Systems.


Therefore, the middle branch of the V − nullcline in somesense separates the firing of an action potential from thesubthreshold return to rest.

This analysis can be made more precise if we assume theparameter φ is small. Looking at the Table shown before,we can see that φ is relatively smaller in the Hopf casethan in the other two cases.



Cable theory II .

Dynamical Systems.


For small φ, n will not change much, so let us hold it atrest. The figure below shows the phase plane with ahorizontal line drawn through the fixed point. If n does notchange much, then the dynamics are governed by thebehavior on the phase line n = nR. Since the V − nullclineintersects this line at three points, there are three equilibriafor the system when n is held constant. The resting state(and true equilibrium of the full system) VR is stable.


Cable theory II .The Morris-Lecar Model. ExcitableSystems



Cable theory II .

Dynamical Systems.


There are two additional equilibria (which are not equilibriaof the full model, just the model when n is held at its restingvalue): Vθ, which is unstable, and Ve, which is stable. Onthis line, if the voltage is perturbed past Vθ, then it will jumpto the right fixed point, Ve. Otherwise, it will decay to rest,VR .

This shows that for small φ, the “threshold” voltage forgenerating an action potential is roughly the intersection ofthe horizontal line through the resting state and the middlebranch of the V − nullcline.



Cable theory II .

Dynamical Systems


Since experimentalists can only move the voltage throughcurrent injection, we can use this to estimate themagnitude of a current pulse needed to cross the threshold

We note that the peak of the action potential occurs atsome latency after the initial perturbation, but this latencycan never become very large



Cable theory II .

Dynamical Systems.


The action potential itself is graded and takes on acontinuum of peak values, as shown in the figure below. Ifφ is not “small” and it is increased, then the spikeamplitudes are even more graded than those shown in thefigure bellow-b


Cable theory II



Cable theory II


Recall that φ is related to the temperature of thepreparation. Thus, increasing the temperature of a neuronshould lead to a much less sharp threshold distinction andgraded action potentials. Indeed, Cole et al. [K. S. Cole, R.Guttman, and F. Bezanilla. Nerve excitation withoutthreshold. Proc. Natl. Acad. Sci., 65:884âAS891, 1970.]demonstrated this in the squid axon.



Cable theory II .

Dynamical Systems.


We expect the phase plane to change if a parameter in theequations changes. in the figure below-d shows the phaseplane corresponding to the periodic solution shown in thethe figure below-b. Here, Iapp = 100.


Cable theory II.

Dynamical Systems.



Cable theory II .

Dynamical Systems.

The Morris-Lecar Model. Oscillations

Note that the periodic solution corresponds to a closedcurve or limit cycle . In general, whenever we wish to findperiodic solutions of some model, we look for closed orbitsin phase space.In figure above-d there is a unique fixedpoint; this is where the nullclines intersect. This fixed pointis unstable, however.



Cable theory II .

Dynamical Systems.


If we change Iapp to 90, then the model is bistable, and thephase plane is shown in the figure bellow-c. Note thatthere exist both a stable fixed point and a stable limit cycle.Small perturbations from rest will decay back to the stablefixed point, whereas large perturbations will approach thestable periodic solution.


Cable theory II.

Dynamical Systems.



Cable theory II .

Dynamical Systems.


Note that there also exists an unstable periodic solution.This orbit separates those initial conditions that approachthe stable fixed point from those that approach the stablelimit cycle.



Cable theory II .

Dynamical Systems.


It is often difficult to show that a given model exhibits stableoscillations, especially in higher-dimensional systems suchas the Hodgkin-Huxley model. Limit cycles are globalobjects, unlike fixed points, which are local.

To demonstrate that a given point is on a periodic solution,one must follow the trajectory passing through that pointand wait to see if the trajectory returns to where it started.This is clearly not a useful strategy for finding periodicsolutions.



Cable theory II .

Dynamical Systems.

The Morris-Lecar Model. Bifurcation Analysis

Bifurcation theory is concerned with how solutions changeas parameters in a model are varied. For example, in theprevious section we showed that the Morris-Lecarequations may exhibit different types of solutions fordifferent values of the applied current Iapp.

If Iapp = 60, then there is a stable fixed point and nooscillations, whereas if Iapp = 100 then the fixed point isunstable and a stable limit cycle exists.



Cable theory II .

Dynamical Systems.

The Morris-Lecar Model. Bifurcation Analysis

Using bifurcation theory, we can classify the types oftransitions that take place as we change parameters. Inparticular, we can predict for which value of I app the fixedpoint loses its stability and oscillations emerge.

There are, in fact, several different types of bifurcations;that is, there are different mechanisms by which stableoscillations emerge.



Cable theory II .

Dynamical Systems.

The Morris-Lecar Model. Bifurcation Analysis-The HopfBifurcation

In the figure bellow, we chose the parameters as in theTable that we showed before, for the Hopf regime andshow the it bifurcation diagram for the Morris-Lecarequations as Iapp is varied.


Cable theory II.

Dynamical Systems.



Cable theory II .

Dynamical Systems.


The curve above the fixed-point curve represents themaximum voltages on the periodic orbits and the curvebelow the fixed-point curve represents the minimumvoltages.

In the figure bellow-a, we plot VR versus Iapp. The fixedpoint is stable for Iapp < 94 ≡ I1 and Iapp > 212 ≡ I2 ;otherwise, it is unstable


Cable theory II.

Dynamical Systems.



Cable theory II .

Dynamical Systems.


A Hopf bifurcation occurs at Iapp = I1 and Iapp = I2. Bythis we mean the following. Recall that a fixed point isstable if all of the eigenvalues of the linearization have anegative real part; the fixed point is unstable if at least oneof the eigenvalues has a positive real part.

The fixed point loses stability, as a parameter is varied,when at least one eigenvalue crosses the imaginary axis.If the eigenvalues are all real numbers, then they can crossthe imaginary axis only at the origin in the complex plane.



Cable theory II .

Dynamical Systems.


However, if an eigenvalue is complex, then it (and itscomplex conjugate) will cross the imaginary axis at somepoint that is not at the origin.

This latter case corresponds to the Hopf bifurcation and itis precisely what happens for the example we areconsidering. In this example, (I1,VR(I1),nR(I1)) and(I2,VR(I2),nR(I2))are called A bifurcation points.Sometimes, I1 and I2 are also referred to as bifurcationpoints.



Cable theory II .

Dynamical Systems.


The Hopf bifurcation theorem* states that (if certaintechnical assumptions are satisfied) there must existvalues of the parameter Iapp near I1 and I2 such that thereexist periodic solutions that lie near the fixed points(VR(Iapp),nR(Iapp)).



Cable theory II .

Dynamical Systems. The Morris-Lecar Model.

Bifurcation Analysis-The Hopf Bifurcation

The Hopf Bifurcation Theorem*

Consider the planar system

x = fµ(x , y)y = gµ(x , y)



Cable theory II .



where µ is a parameter. Suppose it has a fixed point, whichwithout loss of generality we may assume to be located at(x , y) = (0,0). Let the eigenvalues of the linearized systemabout the fixed point be given by λ(µ)(λ(µ)) = α(µ)± iβ(µ).Suppose further that for a certain value of µ (which we mayassume to be 0) the following conditions are satisfied



Cable theory II .

Dynamical Systems. The Morris-Lecar Model.Bifurcation Analysis-The Hopf Bifurcation

α(0) = 0, β(0) = ω 6= 0 wheresgn(w) = sgn[(∂gµ/∂x)|µ0(0,0)]

(non-hyperbolicity condition: conjugate pair of imaginaryeigenvalues))dα(µ)

dµ |µ=0 = d = 6= 0

(transversality condition: the eigenvalues cross theimaginary axis with non-zero speed)

a 6= 0 wherea = 1

16 (fxxx + fxyy + gxxy + gyyy ) +1

ω16 (fxy (fxx + fyy )− gxy (gxx + gyy )− fxxgxx + fyy gyy )

with fxy =(∂2fµ/∂x∂y

)|µ=0(0,0)

(genericity condition)Dr. Marco A Roque Sol Computational Neuroscience. Session 4-3


Cable theory II .

Dynamical Systems.The Morris-Lecar Model.


Then a unique curve of periodic solutions bifurcates from theorigin into the region µ > 0 if ad < 0 or µ < 0 if ad > 0 . Theorigin is a stable fixed point for µ > 0 (resp. µ < 0) and anunstable fixed point for µ < 0 (resp. µ > 0) if d < 0 (resp.d > 0) whilst the periodic solutions are stable (resp. unstable) ifthe origin is unstable (resp. stable) on the side of µ = 0 wherethe periodic solutions exist. The amplitude of the periodic orbitsgrows like

√|µ| whilst their periods tend to 2π/|ω| as |µ| → 0


Cable theory II.Dynamical Systems. The Morris-Lecar Model.


The curves in the figure bellow-a representfixed points and periodic solutions of theMorris-Lecar model.


Cable theory II .

Dynamical Systems.


The curve above the fixed-point curve represents themaximum voltages on the periodic orbits and the curvebelow the fixed-point curve represents the minimumvoltages.

The solid curves represent stable solutions and the dashedcurves represent unstable solutions. The bifurcationdiagram shows many interesting and important features.



Cable theory II .

Dynamical Systems.


Note that the periodic solutions near the two bifurcationpoints are unstable. These unstable, small-amplitudeperiodic solutions lie on the same side of the bifurcationpoints as the stable fixed points. These are both examplesof subcritical Hopf bifurcations.

At a supercritical Hopf bifurcation, the small-amplitudeperiodic solutions near the Hopf bifurcation point are stableand lie on the side opposite the branch of stable fixedpoints.



Cable theory II .

Dynamical Systems.


If 88.3 < Iapp < I1 or I2 < Iapp < 217, then the Morris-Lecarmodel is bistable. For these values of I app , there existboth a stable fixed point and a stable periodic solution.

The phase plane for Iapp = 90 is shown in the figurebellow-c. Note that small perturbations of initial conditionsfrom the resting state will decay back to rest; however,large perturbation from rest will generate solutions thatapproach the stable limit cycles.


Cable theory II.Dynamical Systems. The Morris-Lecar Model.


The figure bellow-b shows the frequencyof the stable periodic solutions versuscurrent.


Cable theory II .

Dynamical Systems.


Note that the frequency lies in a narrow range between 7and 16Hz. In particular, the frequency does not approachzero as Iapp approaches the bifurcation points. This is ageneral property of periodic solutions that arise via theHopf bifurcation.

Finally, in the figure bellow-c we can ask, what happens ifwe change the speed of the potassium kinetics ? Thisfigure shows a two-parameter diagram with φ along thevertical axis and Iapp along the horizontal axis. This showsthe locus of Hopf bifurcations in these two parameters.


Cable theory II.




Cable theory II .

Dynamical Systems.


For fixed values of φ below about 0.4, there are twocurrents at which the Hopf bifurcation occurs.

Inside the curve, the resting state is unstable. One cannumerically show that the Hopf bifurcation is subcriticaloutside the interval 124.47 < Iapp < 165.68. inside thisinterval, the bifurcation is supercritical.



Cable theory II .

Dynamical Systems.

The Morris-Lecar Model. Bifurcation Analysis.Saddle-Node on a Limit Cycle

The Hopf bifurcation is the best known mechanism throughwhich one can go from a stable fixed point to an oscillation.Importantly, the fixed point persists through the bifurcation.

Furthermore, the limit cycles which bifurcate are of smallamplitude and are local, in the sense that they lie close tothe branch of fixed points (although, as we saw in theMorrisâASLecar model, the bifurcation is subcritical at lowcurrents and thus bifurcating periodic orbits are unstable).



Cable theory II .

Dynamical Systems. s


Another mechanism through which an oscillation canemerges from a fixed point is called a saddle-node on alimit cycle (SNLC) . It is also called a saddle-node on aninvariant circle (SNIC). This is an example of a globalbifurcation

The behavior of the Morris-Lecar model with theseparameters is quite different, as is shown in the figurebelow. First, unlike in the figure bellow-b, the actionpotentials appear to occur with arbitrary delay after the endof the stimulus. Second, the shape of the action potentialsis much less variable.


Cable theory II.


Bifurcation Analysis. Saddle-Node on aLimit Cycle


Cable theory II .

Dynamical Systems.


The reason for this can be understood by looking at thephase plane in figure above-b. Unlike the Hopf case, herethere are three fixed points, only one of which (labeled N )is stable. The middle fixed point is a saddle point (labeledS ).

Thus, the linearized system at this fixed point has onepositive and one negative eigenvalue. Associated withthese eigenvalues are the stable and unstable manifolds.



Cable theory II .

Dynamical Systems.


These manifolds consist of trajectories that approach thesaddle point in either forward or backward time,respectively. The two branches of the unstable manifold,Σ+ , form a loop with the stable node N and the saddlepoint S .

This loop in the plane constrains the spike shape; sincetrajectories cannot cross, any trajectory starting outsidethe loop must remain outside it. Thus, the spike heightcannot fall below a certain level



Cable theory II .

Dynamical Systems.


More importantly, the stable manifold, Σ−, forms a hardthreshold that is precisely determined. This contrasts withthe pseudothreshold we saw in the Hopf case.

Any perturbation which drives the potential to the right ofΣ− results in a spike and any perturbation which drives thepotential to the left of Σ− leads to a return to rest without aspike



Cable theory II .

Dynamical Systems.


The figure above also explains the delay in firing. Supposea stimulus drives the voltage to a point exactly on thestable manifold Σ−. Then, the trajectory will go to thesaddle point, where it will remain.

The closer a perturbation gets to Σ− (but to the right of it),the longer the delay to the spike. Indeed, the spike with thelongest delay in Fig. 3.5a stays at a nearly constant voltageclose to the value at the saddle point before finally firing.



Cable theory II .

Dynamical Systems.


Like the Hopf case, as current is increased, the model firesrepetitively. A typical limit cycle is shown in the figurebellow-d.


Cable theory II.




Cable theory II .

Dynamical Systems.


The figure bellow-a shows the bifurcation diagram as thecurrent is increased.The steady-state voltage shows a regionwhere there are three equilibria for Iapp between about −15 and40.


Cable theory II.




Cable theory II .

Dynamical Systems.


Only the lower fixed point is stable. As Iapp increases, thesaddle point and the stable node merge together at asaddle-node bifurcation, labeled SN2. When Iapp = ISN2 ,the invariant l oop formed from Σ+ becomes a homoclinicorbit; that is, it is a single trajectory that approaches asingle fixed point in both forward and backward time

This type of homoclinic orbit is sometimes called asaddleâASnode homoclinic orbit or a SNIC. As Iapp

increases past Iapp = ISN2 , the saddle point and nodedisappear; the invariant loop formed from Σ+ becomes astable limit cycle.



Cable theory II .

Dynamical Systems.


The branch of limit cycles persists until it meets a branch ofunstable periodic solutions emerging from a subcriticalHopf bifurcation.

The figure bellow-b shows the frequency of the oscillationsas a function of the current.


Cable theory II.



Cable theory II .

Dynamical Systems.


Unlike in the figure bellow-b, the frequency forthis model can be arbitrarily low and there is amuch greater dynamic range.


Cable theory II .

Dynamical Systems.


Note that the nullclines in the figure bellow-c can be veryclose to touching each other and thus create a narrowchannel where the flow is extremely slow. This suggestswhy the frequency of firing can be arbitrarily low.


Cable theory II .

Dynamical Systems.



Cable theory II .

Dynamical Systems.


Moreover, as Iapp → ISN2 , the limit cycles approach ahomoclinic orbit. We expect that the frequency shouldapproach zero as Iapp → ISN2



Cable theory II .

Dynamical Systems.

The Morris-Lecar Model. Bifurcation Analysis.Saddle-Homoclinic Bifurcation

By changing the rate of the potassium channel, φ, we canalter the dynamics of the model so that the SNIC isreplaced by another type of global bifurcation; this is calleda saddleâAShomoclinic bifurcation.

In both types of bifurcations, the frequency of oscillationsapproaches zero as the current approaches he bifurcationvalue. However, there are important differences.



Cable theory II .

Dynamical Systems.


Since φ only changes the rate of n, it has no effect on thenumber and values of the fixed points, only their stability.The figure bellow-top shows the bifurcation diagram for themodel when φ is increased from 0.067 to 0.23. As before,the fixed points are lost at a saddle-node bifurcation.


Cable theory II .

Dynamical Systems.




The Hopf bifurcation on the upper branchoccurs at a much lower value of currentthan in the bellow figure but the Hopfbifurcation is still subcritical.


Cable theory II .

Dynamical Systems.


The main difference is that the stable branch of periodicorbits does not terminate on the saddle-node as in figureabove. Rather it terminates on an orbit that is homoclinic toone of the saddle points along the middle branch of fixedpoints.

Like the SNIC, this homoclinic orbit has an infinite period.However, the periods of the limit cycles approach infinityquite differently from before. One can show that the periodscales as:



Cable theory II .

Dynamical Systems.

The Morris-Lecar Model. Bifurcation Analysis.SaddleHomoclinic Bifurcation

T ≈ ln(

1Iapp − IHc

)where IHc is the current at which there is a saddle-homoclinicorbit. The frequency T−1 approaches zero much more rapidlythan in the SNIC case.




The figure bellow-bottom shows the phaseplane for the membrane model near the criticalcurrent, IHc . There are three fixed points.


Cable theory II .

Dynamical Systems.


The lower-left fixed point is always stable, the middle pointis a saddle, and the upper-right point is an unstable spiral.For Iapp < IHc (figure below-bottom-a), the right branch ofthe unstable manifold of the saddle wraps around andreturns to the stable fixed point.


Cable theory II .

Dynamical Systems.



Cable theory II .

Dynamical Systems.


The upper branch of the stable manifold wraps around thespiral (in negative time). Note that the unstable manifoldpasses on the outside of the stable manifold.

In the figure below-bottom-b, the stable and unstablemanifolds meet and form the homoclinic orbit at Iapp = IHc .For Iapp > IHc , the unstable manifold passes inside thestable manifold and wraps around a stable limit cycle.


Cable theory II .

Dynamical Systems.



Cable theory II .

Dynamical Systems.


Thus, this model has a regime of bistability where there isa stable fixed point and a stable periodic orbit. Unlike thebistability in the Hopf case, the stable limit cycle does notsurround the stable fixed point.

In the Hopf case, an unstable periodic orbit acted toseparate the stable fixed point from the stable limitcycle.For the present set of parameters, the stable manifoldof the middle fixed point separates the two stable states.



Cable theory II .

Dynamical Systems.


To get onto the limit cycle, it is necessary to perturb thepotential into the starred region the figure below-bottom-c.Consider a brief current pulse which perturbs the voltage.


Cable theory II .

Dynamical Systems.



Cable theory II .

Dynamical Systems.


If this pulse is weak, then the system returns to rest. If it isvery strong and passes the starred region, then the modelwill generate a single spike and return to rest. However, forintermediate stimuli, the system will settle onto the stablelimit cycle.

Finally, we look closely at the bifurcation diagram (figurebelow-bottom-b). Near I app D 37, there are two stablefixed points as well as a stable limit cycle. Thus, the modelis actually “tristable“ ( three stable states ).


Cable theory II .

Dynamical Systems.



Cable theory II .

Dynamical Systems.

The Morris-Lecar Model. Bifurcation Analysis. Class I andClass II

The Morris-Lecar model illustrates several importantfeatures of neuronal firing.Three different mechanisms forswitching from rest to repetitive firing were illustrated.

In particular, the most common mechanisms are throughthe Hopf and SNIC bifurcations. In the 1940s, Hodgkinclassified three types of axons according to theirproperties.



Cable theory II .

Dynamical Systems.

The Morris-Lecar Model. Bifurcation Analysis.Class I andClass II

He called these classes I and II, with class III beingsomewhere in-between the first two classes which wedescribe:

Class I. Axons have sharp thresholds, can have longlatency to firing, and can fire at arbitrarily low frequencies.

Class II. Axons have variable thresholds, short latency, anda positive minimal frequency.



Cable theory II .

Dynamical Systems.


From this description, we can see that these two classesfall neatly into the dynamics of the SNIC and the Hopfbifurcations, respectively. Rinzel and Ermentrout [J. Rinzeland G. B. Ermentrout. Analysis of neural excitability andoscillations. In C. Koch and I. Segev, editors, Methods inNeuronal Modeling. MIT, Cambridge, MA, 1989.] were thefirst to note this connection.

Now there are many papers which classify membraneproperties as class I or class II and mean SNIC and Hopfbifurcation dynamics, respectively.



Cable theory II .

Dynamical Systems.


Tateno et al. [T. Tateno, A. Harsch, and H. P. Robinson.Threshold firing frequency-current relationships of neuronsin rat somatosensory cortex: type 1 and type 2 dynamics.J. Neurophysiol., 92:2283âAS2294, 2004259 ] havecharacterized regular spiking neurons (excitatory) and fastspiking neurons (inhibitory) in rat somatosensory cortexusing this classification.


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Computational Neuroscience. Session 4-3roquesol/Computational_Neuroscience_Summer... · Session 4-3 Cable theory II . The Morris-Lecar Model. Stability of Fixed Points Dynamical Systems