neuronal excitability from a dynamical systems perspective mike famulare cse/neubeh 528 lecture...

Neuronal excitability from a dynamical systems perspective

Mike FamulareCSE/NEUBEH 528 LectureApril 14, 2009

Mike Famulare, CSE/NEUBEH 528 2

Outline

• Analysis of biophysical models– Hodgkin's classification of neurons by

response to steady input currents

• Introduction to dynamical systems

• Phase portraits and some bifurcation theory– quadratic-integrate-and-fire model

– Fitzhugh-Nagmuo model

• General simple neuron models


Biophysical Modeling

• Neurons can be modelled with a set of nonlinear differential equations (Hodgkin-Huxley)


Hodgkin-Huxley Model


What causes the spike in the HH model?

• In response to a step current at t=5 ms:– fast inward current

followed by slower outward current

– sodium channel (m) activates

– more slowly, potassium channel (n) activates and sodium (h) deinactivates.


Hodgkin-Huxley f-I curve

• Rate coding: firing rate response (f) to input current (I), steady state

• There is a minimum firing rate (58 Hz)• Can you infer the f-I curve from the

Hodgkin-Huxley equations?


Connor-Stevens Model

• Model of a neuron in anisodoris (AKA the sea lemon nudibranch)


What causes a Connor-Stevens spike?


Connor-Stevens f-I curve

• Does not have a minimum firing frequency.


Classifying Neurons by f-I type

cortical pyramidal brainstem mesV

Hodgkin's Classification of Neuronal Excitability

Class 1: shows a continuous f-I curve (like Connor-Stevens)

Class 2: shows a discontinuous f-I curve (like Hodgkin-Huxley)

Class 3: shows no persistent firing (as can be found in auditory brainstem)


Model-by-model is not the way to go!

• We want to understand why neurons are excitable.

• We want to understand what makes different neurons behave in different ways.

• Going model-by-model is difficult and not at all general:– there are hundreds of channel types in nature– any cell expresses a few or ten or so of them

• What do we do with cells whose response is measurable but for which we don't have a model?


Dynamical systems and simple models

• The models we've talked about are dynamical systems.

• What's a dynamical system?

• We can analyze dynamical systems to understand:– equilibria (resting potentials)– “unstable manifolds” (spike thresholds)– bifurcations (f-I class, root of subthreshold

behavior)

v= f v , I


• System: dynamical variables , control variable I.

• Fixed points :• Linear response near a fixed point:

• Stability analysis: what are the eigenvalues of

• Bifurcation: change in the qualitative behavior

of the system as the “control variable”, I, is changed.

Overview of stability and bifurcation analysis

v=f v , I

0=f vo , I

v

vo

v=[∂ f v , I ∂v ]

vo

⋅v−vo

det[[∂ f v , I ∂v ]

vo

−1]=0,stable if ℜ[]0


A note about the leaky-integrate-and-fire

• Leaky-integrate-and-fire (LIF) model:

vo = resting potential, vr = reset voltage, vth = threshold voltage

• This model is not a spiking model in the sense that it doesn't have any dynamics for the spike itself.

• Piecewise “spike” is not dynamically similar to any real neuron

• Useful, but not for what we want to do today.

m v=−v−voIif v≥vth , v vr


Quadratic-Integrate-and-Fire Model (QIF)

• Simplest model with dynamical spikes:m v=−v v2 Iif v≥v s , v v r

v s=spike max , v r=spike reset ,m= time constant ,0


QIF fixed points

• Fixed points:

or

• Do the fixed point exist for all I?

– for , the fixed points no longer exist

(we'll come back to what this means soon!)

0=−v v2I

v+-=1

21±1−4 I

I 14


Phase plane analysis of the QIF: fixed points

• phase portrait for various external currents


Phase plane analysis of the QIF: stability

• Phase portrait of QIF: τm=1, α=1, and Iext=0


Stability Analysis of QIF

• linear response and stability at each fixed point

• for v-

– stable! v- is the resting potential

• for v+

– unstable! v+ is the threshold voltage

m v=[ ddv −v v2]v+-

v−v+-

m v=−21−4 I v−v -

m v=21−4 I v−v +


Saddle-node bifurcation in the QIF

• Loss of stability via a saddle-node bifurcation:– two fixed points “annihilate” each other

• also, we see that the QIF is an integrator!


Two types of saddle-node bifurcation

• “Saddle-node on invariant circle” (SNIC)– reset below “ghost”

of fixed point– arbitrarily low firing

rate—Type I

• Saddle-node (SN)– reset above “ghost”

fp– slow first spike– finite minimum firing

rate—Type II


Summary of saddle-node bifurcations

• Saddle-node bifurcations occur when two fixed points disappear in response to a changing input

• Systems showing an SN bifurcation will be act as integrators

• For neurons, depending on details of the nonlinear spike return mechanism, SN bifurcators can be Type I (continuous f-I curve) or Type II (discontinuous f-I curve)

• The Connor-Stevens model shows a SNIC bifurcation


What about resonating neurons?

• The saddle-node bifurcation type is only one of the very simple (“codimension 1”) bifurcations

• Hodgkin-Huxley does not show a saddle node bifurcation– one of the many ways to see this is that the

HH model cannot show an arbitrarily-long delayed first spike to step current

• With one dynamical variable, the saddle-node is the only possible continuous bifurcation, so we need two variables now!


Fitzhugh-Nagumo Model

• The Fitzhugh-Nagumo (FN) model is Hodgkin-Huxley like.

• Equations:

• Has two dynamical variables (is a two-dimensional dynamical system)– a voltage variable, v

– a recovery variable, w

v=v−v3/3−wIw=a v−bw

a=0.08,b=0.8


Fitzhugh-Nagumo phase portrait

• For standard parameters, the FN has one fixed point that exists for all I.

• dx/dt = 0 lines are known as “nullclines”


Stability of the fixed point in the FN model

• finding the critical current– bifurcation happens when intersection of

nullclines is at the local minimum:

• fixed point nearby critical point

vcrit=−1,wcrit=−1b

I crit=23−1b

I= I− I crit

vo=−1b I O I 2 ,wo=vob


Linear Response of FN near critical point

• Linear response of 2D model:

• eigenvalues are a complex-conjugate pair

• stable when

[∂ f v , I ∂v ]

vo

2 eigenvalues

+-=b I 1± ab

a2b2−4a ±a2b2

4−a

ℜ[+-]=b I

ℜ[+-]0


FN model shows a Hopf bifurcation

• Hopf bifurcation: stable, oscillatory fixed point becomes an unstable, oscillatory fixed point– there is a non-zero minimum firing rate

controlled by the linear response frequency at the critical input current

• Type II excitability

min~ ℑ[+-]

f min~1

2 a− a2b2

4 I

2a b2

4a−a2b2


Visualizing Hopf Dynamics

• phase portrait of a Hopf-bifurcating model (not the FN) for currents below the critical current

• looping around = subthreshold oscillation


Two types of Hopf bifurcation

• There are two types of Hopf bifurcation:– supercritical (like Fitzhugh-Nagumo and HH)

– subcritical

– see “Dynamical Systems in Neuroscience” by Izhikevich or Scholarpedia for details

• For real single neurons, the difference has never been found to be experimentally important (Izhikevich 2007)

• Both are Type II


Bifurcation Theory Review

• Bifurcation: a change in the qualitative behavior in response to a changing control parameter

• With only one control parameter (e.g. current), there are only two types of equilibrium bifurcations (“codimension one”):

• There is a lot more to this bifurcation business!– what if you've got more inputs (drugs,

hormones)?– how do you fit a simple model (“normal form”,

”canonical model”) to a more complex model or real data?


Simple Models can cover a lot of ground

• Saddle-node and Hopf bifurcations are very common and can describe the single-spike properties of the spike-generating mechanisms of most neurons

• One model to do a lot (Izhikevich 2003)

v= v2 vIu=a bv−u

when v≥v spike , v c ,uud


Izhikevich's simple model (adaptive QIF)


References

• Dayan and Abbott

• “Dynamical Systems in Neuroscience” by E.M. Izhikevich

• “Spiking Neuron Models” by Gerstner and Kistler

• “Nonlinear Dynamics and Chaos” by Strogatz

• Scholarpedia

neuronal excitability from a dynamical systems perspective mike famulare cse/neubeh 528 lecture...

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