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Transcript of Basic Models in Theoretical Neuroscience Oren Shriki 2010 Integrate and Fire and Conductance Based...
Basic Models in Theoretical Neuroscience
Oren Shriki
2010
Integrate and Fire and Conductance Based Neurons
1
2
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.
3
The Neuron as an Electric Circuit
4
Intracellular Recording
5
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.
6
Ion Channels and Ion Pumps
7
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:
11
RC circuits• Defining:• We obtain:
• The general solution is:
RC
RtIVdt
dV )(
tIetdeVtVttt
Rt
0
0
12
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
15
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
26
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.
29
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
33
Morris-Lecar Model
• A set of parameters for example:
30
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
34
Morris-Lecar Model
• Voltage dependence of the various parameters (at long times):
35
Conductance-Based Models of Cortical Neurons
36
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.
37
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
38
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?
39
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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