adaptive behaviour research group - kevin...
TRANSCRIPT
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
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
Outline
1 Capturing neural dynamics in the abstract
2 The Fitzhugh-Nagumo model
3 The simple model of Izhikevich
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
Outline
1 Capturing neural dynamics in the abstract
2 The Fitzhugh-Nagumo model
3 The simple model of Izhikevich
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
Outline
1 Capturing neural dynamics in the abstract
2 The Fitzhugh-Nagumo model
3 The simple model of Izhikevich
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
Outline
1 Capturing neural dynamics in the abstract
2 The Fitzhugh-Nagumo model
3 The simple model of Izhikevich
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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)
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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)
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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 )
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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 )
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
Outline
1 Capturing neural dynamics in the abstract
2 The Fitzhugh-Nagumo model
3 The simple model of Izhikevich
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
Outline
1 Capturing neural dynamics in the abstract
2 The Fitzhugh-Nagumo model
3 The simple model of Izhikevich
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
Dynamics with no input
Nullclines and key features of Izhikevich simplemodel with I = 0
Note that the parametershave been chosen so thatthe membrane potential maybe plausibly interpreted inmV
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
Appearance of spikes
Izhikevich simple model with I = 100
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
Appearance of spikes
Izhikevich simple model with I = 100
In the time domain, it isclear that there is repetitivespiking
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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))
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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.
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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.
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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.
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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.
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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
K.N. Gurney PSY6308: Single neuron models
OutlineCapturing neural dynamics in the abstract
The Fitzhugh-Nagumo modelThe simple model of Izhikevich
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)
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K.N. Gurney PSY6308: Single neuron models