Basic Models in Theoretical Neuroscience
Oren Shriki
2010
Integrate and Fire and Conductance Based Neurons
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References about neurons as electrical circuits:
• Koch, C. Biophysics of Computation, Oxford Univ. Press, 1998.
• Tuckwell, HC. Introduction to Theoretical Neurobiology, I&II, Cambridge UP, 1988.
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The Neuron as an Electric Circuit
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Intracellular Recording
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Generation of Electric Potential on Nerve Cell Membranes
Chief factors that determine the resting membrane potential:
• The relative permeability of the membrane to different ions
• Differences in ionic concentrations
Ion pumps – Maintain the concentration gradient by actively moving ions against the gradient using metabolic resources.
Ion channels – “Holes” that allow the passage of ions in the direction of the concentration gradient. Some channels are selective for specific ions and some are not selective.
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Ion Channels and Ion Pumps
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The Neuron as an Electric Circuit
• Differences in ionic concentrations Battery
• Cell membrane Capacitor
• Ionic channels Resistors
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The Neuron as an Electric Circuit
Extracellular
Intracellular9
RC circuits
• R – Resistance (in Ohms)
• C – Capacitance (in Farads)
I RCCurrent
source
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RC circuits
I RC
)(tIR
V
dt
dVC
• The dynamical equation is:
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RC circuits• Defining:• We obtain:
• The general solution is:
RC
RtIVdt
dV )(
tIetdeVtVttt
Rt
0
0
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RC circuit• Response to a step current:
0 0
0
0
0 0
( | 1 1
0 1
t tt tt t
t t t t tt
t t
V
I tI t
t
dt e I t e I dt e
e I e e I e I e
V t V e IR e
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RC circuit• Response to a step current:
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The Integrate-and-Fire Neuron
• R – Membrane Resistance (1/conductance)
• C – Membrane Capacitance (in Farads)
I RC
inside
outside
EL
Threshold mechanism
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Integrate-and-Fire Neuron
• If we define:
• The dynamical equation will be:
• To simplify, we define:
• Thus:
outin VVV
)(1
tIEVRdt
dVC L
LEVVV outin
)(tIR
V
dt
dVC
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Integrate-and-Fire Neuron
• The threshold mechanism:
– For V<θ the cell obeys its passive dynamics– For V=θ the cell fires a spike and the voltage resets to
0.
• After voltage reset there is a refractory period, τR.
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Integrate-and-Fire Neuron• Response to a step current:
IR<θ:
t
V
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Integrate-and-Fire Neuron
• Response to a step current: IR>θ:
V
t
T
τR τR τR
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Integrate-and-Fire Neuron
• Finding the firing rate as a function of the applied current:
1
1 1
1 1
1
R
R R
T
T T
RR
R
V t IR e
e eIR IR
Tln T ln
IR IR
T lnIR
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Integrate-and-Fire Neuron 1 1 1
1 1
1
1
R R
LR
L
f IT
ln lnIR IR
gCln
g I
f
I
1
R
cIR
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The Hodgkin-Huxley Equations
n
h
m
/τ(V)-nndn/dt
/τ(V)-hhdh/dt
/τ(V)-mmdm/dt
)(, tIwVIdt
dVC ion
)()()(
,,,
LLK4
KNa3
Na VVgVVngVVhmg
nhmVI ion
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The Hodgkin & Huxley Framework
)(,,, 1 tIWWVIdt
dVC Nion
V
WVW
dt
dW
i
iii
,
Each gating variable obeys the following dynamics:
i
- Represents the effect of temperature
- Time constant
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The Hodgkin & Huxley Framework
)(,,, 1 jNjjj VVWWVgI
The current through each channel has the form:
j
jg - Maximal conductance (when all channels are open)
- Fraction of open channels (can depend on several W variables). 24
The Temperature Parameter Φ
• Allows for taking into account different temperatures.
• Increasing the temperature accelerates the kinetics of the underlying processes.
• However, increasing the temperature does not necessarily increase the excitability. Both increasing and decreasing the temperature can cause the neuron to stop firing.
• A phenomenological model for Φ is:
10/3.6Temp3 25
Hodgkin & Huxley Model
n
h
m
/τ(V)-nndn/dt
/τ(V)-hhdh/dt
/τ(V)-mmdm/dt
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Ionic Conductances During an Action Potential
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Repetitive Firing in Hodgkin–Huxley Model
A: Voltage time courses in response to a step of constant depolarizing current. from bottom to top: Iapp= 5, 15, 50, 100, 200 in μamp/cm2). Scale bar is 10 msec. B: f-I curves for temperatures of 6.3,18.5, 26◦C, as marked. Dotted curves show frequency of the unstable periodic orbits. 28
Fast-Slow Dissection of the Action Potential
• n and h are slow compared to m and V.• Based on this observation, the system can be
dissected into two time-scales.• This simplifies the analysis.• For details see:
Borisyuk A & Rinzel J. Understanding neuronal dynamics by geometrical dissection of minimal models. In, Chow et al, eds: Models and Methods in Neurophysics (Les Houches Summer School 2003), Elsevier, 2005: 19-72.
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Correlation between n and h
• During the action potential the variables n and h vary together.
• Using this correlation one can construct a reduced model.
• The first to observe this was Fitzhugh.
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Simplified Versions of the HH Model
• Models that generate action potentials can be constructed with fewer dynamic variables.
• These models are more amenable for analysis and are useful for learning the basic principles of neuronal excitability.
• We will focus on the model developed by Morris and Lecar.
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The Morris-Lecar Model (1981)
• Developed for studying the barnacle muscle.
• Model equations:
)(, tIwVIdt
dVC appion
V
wVw
dt
dw
w
)()())((, LLKKCaCa VVgVVwgVVVmgwVI ion 32
Morris-Lecar Model• The model contains K and Ca currents.• The variable w represents the fraction of open K
channels.• The Ca conductance is assumed to behave in an
instantaneous manner.
21 /tanh15.0)( VVVVm
43 2/cosh/1 VVVVw
43 /tanh15.0)( VVVVw
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Morris-Lecar Model
• A set of parameters for example:
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2
18
2.1
4
3
2
1
V
V
V
V
2
8
4
L
K
Ca
g
g
g
60
84
120
L
K
Ca
V
V
V
04.0cm
μF20
2
C
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Morris-Lecar Model
• Voltage dependence of the various parameters (at long times):
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Conductance-Based Models of Cortical Neurons
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Conductance-Based Models of Cortical Neurons
• Cortical neurons behave differently than the squid axon that Hodgkin and Huxley investigated.
• Over the years, people developed several variations of the HH model that are more appropriate for describing cortical neurons.
• We will now see an example of a simple model which will later be useful in network simulations.
• The model was developed by playing with the parameters such that its f-I curve is similar to that of cortical neurons.
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Frequency-Current Responses of Cortical Neurons
Excitatory Neuron:
Ahmed et. al., Cerebral Cortex 8, 462-476, 1998
Inhibitory Neurons:
Azouz et. al., Cerebral Cortex 7, 534-545, 1997
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Frequency-Current Responses of Cortical Neurons
The experimental findings show what f-I curves of cortical neurons are:
• Continuous – starting from zero frequency.• Semi-Linear – above the threshold current the
curve is linear on a wide range.
How can we reconstruct this behavior in a model?
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An HH Neuron with a Linear f-I Curve
Shriki et al., Neural Computation 15, 1809–1841 (2003) 40
Linearization of the f-I Curve
• We start with an HH neuron that has a continuous f-I curve (type I, saddle-node bifurcation).
• The linearization is made possible by the addition of a certain K-current called A-current.
• The curve becomes linear only when the time constant of the A-current is slow enough (~20 msec).
• There are other mechanisms for linearizing f-I curves.
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Model Equations:
Shriki et al., Neural Computation 15, 1809–1841 (2003)
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