Adaptive Behaviour Research Group - Kevin...

OutlineCapturing neural dynamics in the abstract

The Fitzhugh-Nagumo modelThe simple model of Izhikevich

Single neuron modelsReduced models and phase-plane analysis of their dynamics: 3

Kevin Gurney

Adaptive Behaviour Research Group

2008

K.N. Gurney PSY6308: Single neuron models



Outline

1 Capturing neural dynamics in the abstract

2 The Fitzhugh-Nagumo model

3 The simple model of Izhikevich




Outline







Outline







Outline







Capturing the essentials of spiking dynamics

We know the essence of the dynamic behaviour is captured bythe nullclines - their functional forms and their intersection atequilibria

The nullclines act as ‘guidelines’ that lend structure to thevector field which, in turn determines trajectories

Hitherto, these functional forms have arisen fromphysiologically plausible models and the Hodgkin-Huxleyformalism

A new strategy is to establish models, independent of anyphysiological framework, whose dynamics are equivalent totheir biologically grounded counterparts

This programme requires we describe, as simply as possible,the nullclines of typical neural models








































The V -nullcline

The V -nullcline of bothmodels we have studiedconsists of an inverted‘N’-shape

The simplest function of Vwhich has such a form is

y(V ) = V − AV 3 + B (1)

i.e. a simple ‘cubic’ withparameters A, B




The V -nullcline

The V -nullcline of bothmodels we have studiedconsists of an inverted‘N’-shape

The simplest function of Vwhich has such a form is

y(V ) = V − AV 3 + B (1)

i.e. a simple ‘cubic’ withparameters A, B




The V -nullcline

If the recovery variable in our model is w , then a membraneequation of the form

dV

dt= V − AV 3 − w + I (2)

gives rise to a nullcline of the form in (1), for puttingdV /dt = 0 gives

w = V − AV 3 + I (3)

where I ↔ B in (1)

calling the recovery variable w , helps remind us it is an abstract variable and

not related to the HH-formalism like n




The w -nullcline

In the reduced-HH model, thenullcline for the recovery variable isroughly linear over the relevantpart of phase space

if we aim to capture HH model-likedynamics, then we may thereforeapproximate the nullcline by alinear function

w = CV + D (4)




The w -nullcline

In the reduced-HH model, thenullcline for the recovery variable isroughly linear over the relevantpart of phase space

if we aim to capture HH model-likedynamics, then we may thereforeapproximate the nullcline by alinear function

w = CV + D (4)




The w -nullcline

In fact, a linear w -nullcline issufficient to enable many variationsof the dynamics

The number, and location, ofintersections with the V -nullcline isdetermined by the slope of thew -nullcline, and these are key tothe dynamics




The w -nullcline

In fact, a linear w -nullcline issufficient to enable many variationsof the dynamics

The number, and location, ofintersections with the V -nullcline isdetermined by the slope of thew -nullcline, and these are key tothe dynamics




The w -nullcline

A recovery variable equation of the form

dw

dt= CV + D − w (5)

gives rise to a w -nullcline of the form in (4) (put dw/dt = 0)

(5) may be written

dw

dt= C (V + E − Fw) (6)

where the new constants E , F are given in terms of the oldones by E = D/C , F = 1/C )




The w -nullcline

A recovery variable equation of the form

dw

dt= CV + D − w (5)

gives rise to a w -nullcline of the form in (4) (put dw/dt = 0)

(5) may be written

dw

dt= C (V + E − Fw) (6)

where the new constants E , F are given in terms of the oldones by E = D/C , F = 1/C )




Outline







The Fitzhugh-Nagumo model

Equations (2) and (6) constitute the model developed byFitzhugh (1961)

In particular, one parametrisation which gives dynamicscharacteristic of an HH-like model is

An HH-like model

dV

dt= V − V 3

3− w + I (7)

dw

dt= 0.8(V + 0.7− 0.8w)

An implementation of a similar model in electronic hardwareby Nagumo in 1962 leads to the joint namingFitzhugh-Nagumo for this model







An HH-like model

dV

dt= V − V 3

3− w + I (7)

dw

dt= 0.8(V + 0.7− 0.8w)








An HH-like model

dV

dt= V − V 3

3− w + I (7)

dw

dt= 0.8(V + 0.7− 0.8w)





Phase diagram and resting state

The nullclines with I = 0 areexactly as we expect (theywere constructed to be so!)

A typical trajectoryterminating on a a stablefocus is shown




Phase diagram and resting state

The nullclines with I = 0 areexactly as we expect (theywere constructed to be so!)

A typical trajectoryterminating on a a stablefocus is shown




Status of variables and constants

The variables V and w are supposed to correspond to themembrane potential and recovery variable respectively

However, the model pays no heed to dimensions, so a value of1.5, say, for V should not be interpreted as 2mV or 2V

Physiologically plausible units could be imposed by rescalingall the constants


















Limit cycles and spikes

With a ‘current’ of 0.33, regular spiking is observed associatedwith a limit cycle

The model displays the same bifurcations as the reduced HHmodel and thereby captures the essentials of its dynamics




Limit cycles and spikes

With a ‘current’ of 0.33, regular spiking is observed associatedwith a limit cycle

The model displays the same bifurcations as the reduced HHmodel and thereby captures the essentials of its dynamics




Equivalent forms

Suppose we chose to use the variable V̂ = V − Vrest

Substituting V = V̂ + Vrest into (7) would give a new set ofequations in the new membrane potential variable V̂

However, the membrane equation would still be a cubic in V̂(albeit with quadratic terms too), and the recovery variableequation would still be affine in V̂

Variable substitutions lead to equivalent forms for the modelwhich involve cubics in the membrane equation and linear (oraffine) expressions in the recovery equation

These would all be classified as Fitzhugh-Nagumo models




Equivalent forms









Equivalent forms









Equivalent forms









Equivalent forms









Outline







Further simplification

The Fitzhugh-Nagumo model is a considerable computationalsimplification compared to the physiological models

The latter require expensive computations such as sigmoidfunctions for gating variables

Is it possible to simplify further?

Computation is most intensive when a system is changingmost rapidly - time must be partitioned very finely to ensurethe correct dynamics are computed

For neuron models, this corresponds to computing thebehaviour during a spike

Izhikevich has recently proposed a simple 2D model of neuralbehaviour which circumvents spike computation but which isrich enough to model a wide variety of spiking behaviours












































Spike initiation is localised in phase space

spike initiation in the Na+p − K+ model

In all the reduced 2D modelswe have seen, the overallspiking behaviour iscontrolled in a fairly limitedpart of phase space (Rinit)around the ‘dip’ in theV -nullcline

The vector field heredetermines if a spike will beinitiated, or if the trajectoryreturns to a stableequilibrium




Spike initiation is localised in phase space

spike initiation in the Na+p − K+ model

In all the reduced 2D modelswe have seen, the overallspiking behaviour iscontrolled in a fairly limitedpart of phase space (Rinit)around the ‘dip’ in theV -nullcline

The vector field heredetermines if a spike will beinitiated, or if the trajectoryreturns to a stableequilibrium




Spike recovery is localised in phase space

spike recovery in the Na+p − K+ model

Similarly, the recovery phase(or ‘downstroke’) of thespike is controlled by therightmost portions (Rrec) ofthe V -nullcline




Bypassing spike simulation per se

Izhikevich suggested that we could bypass the computationallyintensive process of spike simulation as follows

1 Simulate the subthreshold behaviour of the model in limitedphase space like that in Rinit

2 Watch for spike initiation by noting if V reaches some valueVpeak

3 If a spike is detected, then reset the membrane and recoveryvariables to values back in Rinit

We now quantify this scheme








































Describing nullclines in Rinit

The V -nullcline in a regionlike Rinit appears to beroughly quadratic

The recovery variablenullcline appears to beroughly linear

We uses these descriptionsto define the simple model


















The Izhikevich simple model

The simple model: membrane potential V , recovery variable u

dV

dt=

1

C[k(V − Vr )(V − Vt)− u + I ] (8)

du

dt= a [b(V − Vr )− u] (9)

if V ≥ Vpeak then V ← c , u ← u + d (10)

Here, C behaves like the membrane capacitance, Vr is the restingpotential, I the injection current. Other quantities are illustrated inthe next two slides






dV

dt=

1

C[k(V − Vr )(V − Vt)− u + I ] (8)

du

dt= a [b(V − Vr )− u] (9)








dV

dt=

1

C[k(V − Vr )(V − Vt)− u + I ] (8)

du

dt= a [b(V − Vr )− u] (9)






Dynamics with no input

Nullclines and key features of Izhikevich simplemodel with I = 0

Some features with I = 0 are:

In the resting state, u = 0

Vt , Vr are the values of Vwhen u = 0

The slope of the n-nullclineis the constant b in (8)

With the parameters chosenhere, there is a stable nodeequilibrium at Vr and asaddle node equilibrium atVs > Vt




































Note that the parametershave been chosen so thatthe membrane potential maybe plausibly interpreted inmV




Appearance of spikes

Izhikevich simple model with I = 100

There are no equilibria atI = 100 and so spiking is anecessity

The trajectory rapidly givesrise to the up-phase of aspike (shown in red)

This would carry on toindefinitely large values of Vunless reset at Vpeak






















While the reset potential isalways c , the reset value ofu is u + d , which may vary(the initial spike has asmaller value of u at reset)

The value of c is at theminimum of theV -nullcline here - this ispurely incidental






In the time domain, it isclear that there is repetitivespiking




LIF and simple model compared

In the LIF model, spikes are initiated at a threshold

In the simple model they are initiated by the dynamics of thesystem (as in the more complex reduced models) andterminated at a peak value




LIF and simple model compared

In the LIF model, spikes are initiated at a threshold

In the simple model they are initiated by the dynamics of thesystem (as in the more complex reduced models) andterminated at a peak value




The simple model is a powerful one

The simple model is capableof a very wide diversity offiring behaviour usingdifferent paramaterisations[figure from ch 8 of(Izhikevich, 2007))




Summary

It is possible to abstract the essential dynamics of a spikingneuron by establishing equations that give neural-like phaseplane structures (nullclines and vector fields)

These models have, however, abandoned direct links with anyphysiological basis

This strategy gives the Fitzhugh-Nagumo model and thesimple models of Izhikevich

In sum, the analysis of 2D, reduced models is a very powerfulapproach giving insights into a range of phenomena including,excitability classification, rebound inhibition etc.




Summary








Summary








Summary








Further reading

Izhikevich and Fitzhugh (2006) have an article on theFitzhugh-Nagumo model in Scholarpedia which has a nicehistorical note

Ch 8 of the book by Izhikevich (2007) deals with the SimpleModel

Matlab code is available on MOLE




References

Izhikevich, E. (2007). Dynamical systems in neuroscience: The geometry of excitability and bursting. MITPress.

Izhikevich, E., & Fitzhugh, R. (2006). Fitzhugh-nagumo model. In Scholarpedia (p. 5642). online material.(http://www.scholarpedia.org/wiki/index.php?title=FitzHugh-Nagumo Model&&oldid=7128)

1


Date post:	25-Apr-2018
Category:	Documents
Upload:	lamthien
View:	219 times
Download:	2 times

Adaptive Behaviour Research Group - Kevin...

Documents