Neuronal excitability from a dynamical
systems perspective
Mike Famulare
CSE/NEUBEH 528 Lecture
April 14, 2009
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
Mike Famulare, CSE/NEUBEH 528
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Biophysical Modeling
• Neurons can be modelled with a set of nonlinear
differential equations (Hodgkin-Huxley)
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Hodgkin-Huxley Model
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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.
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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 HodgkinHuxley equations?
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Connor-Stevens Model
• Model of a neuron in anisodoris (AKA the sea
lemon nudibranch)
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What causes a Connor-Stevens spike?
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Connor-Stevens f-I curve
• Does not have a minimum firing frequency.
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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)
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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?
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Dynamical systems and simple models
• The models we've talked about are dynamical
systems.
• What's a dynamical system?
v̇= f v , I
• We can analyze dynamical systems to understand:
– equilibria (resting potentials)
– “unstable manifolds” (spike thresholds)
– bifurcations (f-I class, root of subthreshold behavior)
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⋅ℜ
Overview of stability and bifurcation analysis
• System: dynamical variables v , control variable I.
v̇= f v , I
• Fixed points v o :
0= f v , I
o
• Linear response near a fixed point:
∂ f v,I
v̇=
v − vo
∂v
v
• Stability analysis: what are the eigenvalues of
[
[[
]
] ]
o
∂ f v,I
det
− 1 = 0,stable if [ ] 0
∂v
v
• Bifurcation: change in the qualitative behavior of
the system as the “control variable”, I, is
changed.
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o
A note about the leaky-integrate-and-fire
• Leaky-integrate-and-fire (LIF) model:
v̇= − v− vo I
if v≥ v th , v v r
m
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.
Mike Famulare, CSE/NEUBEH 528
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Quadratic-Integrate-and-Fire Model (QIF)
• Simplest model with dynamical spikes:
m v̇= −
v
if v≥ v s , v
v
vr
2
I
v s = spike max , v r = spike reset ,
0
m = time constant ,
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QIF fixed points
• Fixed points:
0= − v
v I
1
v +-=
1± 1− 4 I
2
2
or
• Do the fixed point exist for all I?
– for I
1 , the fixed points no longer exist
4
(we'll come back to what this means soon!)
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Phase plane analysis of the QIF: fixed points
• phase portrait for various external currents
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Phase plane analysis of the QIF: stability
• Phase portrait of QIF: τm=1, α=1, and Iext=0
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Stability Analysis of QIF
• linear response and stability at each fixed point
[
d
−v
m v̇=
dv
v
2
]
v− v +-
v +-
• for vm
v̇= − 2
1− 4 I v− v -
– stable! v- is the resting potential
• for v+
– unstable!
v 2is the
1− threshold
4 I v− vvoltage
m v̇= +
+
Mike Famulare, CSE/NEUBEH 528
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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!
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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
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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
Mike Famulare, CSE/NEUBEH 528
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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!
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Fitzhugh-Nagumo Model
• The Fitzhugh-Nagumo (FN) model is HodgkinHuxley like.
3
v̇=
v−
v
/3− w I
• Equations:
ẇ= a v− b w
a= 0.08, b= 0.8
• Has two dynamical variables (is a twodimensional dynamical system)
– a voltage variable, v
– a recovery variable, w
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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”
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Stability of the fixed point in the FN model
• finding the critical current
– bifurcation happens when intersection of nullclines
is at the local minimum:
−1
v crit = − 1, w crit =
b
2 1
I crit = −
3 b
• fixed point nearby critical point
I = I − I crit
v o= − 1 b
I
O
I
2
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vo
, w o=
b
26
ℜ
Linear Response of FN near critical point
• Linear response of 2D model:
[
]
∂ f v ,I
∂v
+-
=b
2 eigenvalues
vo
ab
a 2 b2
±
−
a
4
a 2 b 2− 4a
I 1±
• eigenvalues are a complex-conjugate pair
[
• stable when
[
+-
+-
]= b
I
] 0
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ℑ
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
min
1
f min ~
2
~ [
2
a b
a−
4
• Type II excitability
+-
2
]
I
2
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2
ab
2 2
4a− a b
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Visualizing Hopf Dynamics
• phase portrait of a Hopf-bifurcating model (not
the FN) for currents below the critical current
• looping around = subthreshold oscillation
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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
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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?
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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)
v̇= v 2
v
u̇= a bv− u
when v≥ v spike , v
I
c ,u
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u d
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Izhikevich's simple model (adaptive QIF)
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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
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