JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models
-
Upload
hirokazutanaka -
Category
Education
-
view
63 -
download
1
Transcript of JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models
![Page 1: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/1.jpg)
SS2016 Modern Neural Computation
Lecture 1: Single Neurons
Hirokazu TanakaSchool of Information Science
Japan Institute of Science and Technology
![Page 2: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/2.jpg)
Neuron as a computational unit of the brain.
In this lecture we will learn:• Basic anatomy and physiology of neuron
- morphology- membrane properties
• Phenomenological models with subthreshold dynamics- Integrate-and-fire model, Quadratic-and-fire model, Resonate-and-fire model
• Biophysical models with spiking mechanism- Ion channels, master equations- Hodgkin-Huxley model
• Phase plots and bifurcation analysis- Saddle-node bifurcation, Andronov-Hopf bifurcation- FitzHugh-Nagumo model, Hindmarsh-Rose model
• Modern single-neuron models- Izhikevich model, Adaptive-exponential model
![Page 3: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/3.jpg)
Neurons composed of dendrites, soma and axon.
Figure 3.1, Fundamental Neuroscience, 3rd Edition
![Page 4: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/4.jpg)
Morphology: Neurons take various shapes.
Figure 2.1, Fundamental of Computational Neuroscience
![Page 5: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/5.jpg)
Cortical neurons receive cortico-cortical and thalamo-cortical inputs.
Figure 3.2, Fundamental Neuroscience, 3rd Edition
Pyramidal cell in layer II/III
Apical dendrites
Basal dendrites
![Page 6: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/6.jpg)
Lipid-bilayer membrane insulates a neuron
Ruye Wang, http://fourier.eng.hmc.edu/e180/lectures/signal1/node2.html
![Page 7: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/7.jpg)
Physiology: Neurons are electrically excitable.
Figure 2.2, Neuroscience 3rd Edition
![Page 8: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/8.jpg)
Physiology: Neurons take various spiking patterns.
Izhikevich (2004) IEEE Neural Networks
![Page 9: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/9.jpg)
Leaky Integrate-and-fire model (LIF)
( ) ( )( ) ( )m L
dv tv t E RI t
dtτ = − − +
( ) ( )f
f
j
j jj t
I t w t tα= −∑∑
( )fthv t V=
( )freset .v t V←
Leaky integration Fire (spike)If the potential reaches the threshold voltage,
then, add a spike and reset the potential to the reset voltage.
Figure 3.1, Fundamental of Computational Neuroscience
Lapicque (1907)For English translation, see:
Brunel & van Rossum (2007)
![Page 10: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/10.jpg)
Analytical solution of LIF model with constant current.
( ) ( )m
dv tv t RI
dtτ = − +
( ) ( ) m m0 1 t t
tv t v e RI e RIτ τ
− −
→∞
= + − →
Figure 3.2, Fundamental of Computational Neuroscience
( ) ( )( ) const.
Lv t v t E
I t I
← −
= =
subtracting the equilibrium potential.
considering a time-invariant current.
![Page 11: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/11.jpg)
f-I curve of LIF model.
Figure 3.3, Fundamental of Computational Neuroscience
( )L r
ref mL th
1
lnf I
RI E VRI E V
τ τ=
+ −+ + −
![Page 12: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/12.jpg)
Quadratic-and-fire model
( ) ( )2dv tI v t
dt= +
( ) ( )thresholdif , then resetv t v v t v≥ ←
( ) ( , )dv t
F v Idt
=
In general, the dynamics for membrane potential has a general form:
Quadratic-and-fire (QIF) model: F is quadratic in terms of v and linear in terms of I.
For LIF model, F is linear in terms of both v and I.
( , )F v I v I= − +
![Page 13: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/13.jpg)
Quadratic-and-fire model
Figure 3.35, Dynamical Systems in Neuroscience
![Page 14: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/14.jpg)
Resonate-and-fire model: oscillatory sub-threshold dynamics.
( ) ( )( ) ( )
( ) ( )( ) ( )
leak leak
1/2
dv tC I g v t E w t
dtv t vdw t
w tdt k
= − − −
−= −
For some neurons, the sub-threshold dynamics exhibits an oscillatory behavior:
Resonate-and-fire model: two-dimensional model of membrane potential (v) and the recovery variable (w).
Whole-cell recording of an olivary neuron
Hutcheon & Yarom (2000) Trends Neurosci
![Page 15: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/15.jpg)
Resonate-and-fire model.
Izhikevich (2001) Neural Networks
spike
spikeno spike
no spike
![Page 16: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/16.jpg)
Ion channels: Nernst equation.
Figure 6.3, Fundamental of Computational Neuroscience
[ ][ ]
oution in out
in
ionln
ionRTE E EzF
≡ − =
EoutEin Nernst equation
[ ][ ]
( )out
out in
in
out
in
ionion
zF E zFRT E ERT
zF ERT
e ee
−− −
−
= =
in[ ] 140mMK + = out[ ] 3mMK + =[ ][ ]
out
in
3ln 61.5ln 102mV140K
KRTEF K
= = = −
Potassium ion
![Page 17: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/17.jpg)
Ion channels: Goldman-Hodgkin-Katz equation.
Figure 6.3, Fundamental of Computational Neuroscience
K Na Clout out inm out in
K Na Clin in out
K Na Clln
K Na Cl
p p pRTV V VF p p p
+ + −
+ + −
+ + = − = + +
Goldman-Hodgkin-Katz equation
K Na Cl: : 1.00 : 0.04 : 0.45p p p =
Permeability
For T=293K (20°C), the equilibrium potential is
m out in 62mVV V V= − = −
![Page 18: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/18.jpg)
Ion-channel kinetics: voltage-dependent ion channels
: activation variablen
( )( ) ( )1n ndn V n V ndt
α β= − −Inactive Active
( )n Vα
( )n Vβ
( )activeP n=( )inactive 1P n= −
Master equation
( ) ( )ndnV n V ndt
τ ∞= −
( ) ( ) ( )1
mn n
VV V
τα β
=+
( ) ( )( ) ( )
n
n n
Vn V
V Vα
α β∞ =+
time constant
asymptotic value
Gating equation
![Page 19: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/19.jpg)
Hodgkin-Huxley model: potential and gating dynamics.
( ) ( ) ( )4 3K K Na Na L L
dVC g n E V g m h E V g E V Idt
= − − − − − − +
( ) ( )ndnV n V ndt
τ ∞= −
( ) ( )mdmV m V mdt
τ ∞= −
( ) ( )hdhV h V hdt
τ ∞= −
Membrane-potential dynamics
Gating equations
Figure 5.10, Theoretical Neuroscience
![Page 20: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/20.jpg)
Hodgkin-Huxley model: activation and inactivation variables.
( ) ( ) ( )4 3K K Na Na L L
dVC g n E V g m h E V g E V Idt
= − − − − − − +
( ) ( )ndnV n V ndt
τ ∞= −
( ) ( )mdmV m V mdt
τ ∞= −
( ) ( )hdhV h V hdt
τ ∞= −
Membrane-potential dynamics
Gating equations m: Na+ activation variableh: Na+ inactivation variablen: K+ activation variable
Figure 2.8, Dynamical Systems in Neuroscience
m=0h=1
m=1h=1
m=1h=0
![Page 21: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/21.jpg)
Hodgkin-Huxley model reproduces spike waveform.
Figure 5.10, Theoretical Neuroscience Figure 4.3, Neuroscience 3rd Edition
![Page 22: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/22.jpg)
Hodgkin-Huxley model reproduces spike waveform.
Figure 2.15, Dynamical Systems in Neuroscience
![Page 23: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/23.jpg)
Phase-plane plot: one-dimensional case
( ),dV F V Idt
=
*( , ) 0 fixed pointF V I = →( )( )
*
*
, 0 stable (attractive) fixed point
, 0 unstable (repulsive) fixed point
F V I
F V I
′ < →
′ > →
Figure 3.10, Dynamical Systems in Neuroscience
Phase-plane plot: schematic method for capturing qualitative behaviors of differential equations without solving.
Figure 3.18, Dynamical Systems in Neuroscience
![Page 24: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/24.jpg)
Bifurcation: Saddle-node bifurcation
( )dV F V Idt
= +
Figure 3.25, Dynamical Systems in Neuroscience
![Page 25: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/25.jpg)
Phase-plane plot: two-dimensional case
( )( )
,,
V F V ww G V w = =
Phase-plane plot: vector field (dV/dt, dw/dt) on the two dimensional plane.
Figure 4.3, Dynamical Systems in Neuroscience
1, 0x y= = 0, 1x y= =
, x x y y= − = − , x y y x= − = −
![Page 26: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/26.jpg)
Phase-plane plot: Nullclines
( )( )
,,
V F V ww G V w = =
Nullclines: the curves of F(V,w)=0 and G(V,w)=0.
Figure 4.3, Dynamical Systems in Neuroscience
![Page 27: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/27.jpg)
Phase-plane plot: linear stability analysis
( )( )
,,
V F V ww G V w = =
Phase-plane plot: vector field (dV/dt, dw/dt) on the two dimensional plane.
Dynamical Systems with Applications using MATLAB
Stable node Unstable node Saddle point
Unstable focus Stable focus Center
![Page 28: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/28.jpg)
Phase-plane plot: Separatrix
( )( )
,,
V F V ww G V w = =
Phase-plane plot: vector field (dV/dt, dw/dt) on the two dimensional plane.
Figure 4.24, Dynamical Systems in Neuroscience
Separatrix: the boundary separating two modes of behaviour in a differential equation.
![Page 29: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/29.jpg)
Bifurcation: Saddle-node bifurcation
Figure 4.26, 28, 30, Dynamical Systems in Neuroscience
![Page 30: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/30.jpg)
Bifurcation: (supercritical) Andronov-Hopf bifurcation
Figure 4.26, 28, 30, Dynamical Systems in Neuroscience
![Page 31: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/31.jpg)
Class I and II neurons and bifurcation type
Figure 7.3, Dynamical Systems in Neuroscience
Class I: Continuous F-I curve, Saddle-node bifurcationClass II: Discontinuous F-I curve, Andronov-Hopf bifurcation
![Page 32: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/32.jpg)
Two-dim. model: FitzHugh-Nagumo model
( )
3
30.08 0.8 0.7
vv v w I
w v w
= − − +
= − +
FitzHugh (1961) Biophysical J; “FitzHugh-Nagumo Model” (2015) Encyclopedia of Comp Neuro
stable unstable
*I I< *I I<
![Page 33: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/33.jpg)
Two-dim. model: FitzHugh-Nagumo model
( )3
, 0.08 0.8 0.73vv v w I w v w= − − + = − +
FitzHugh (1961) Biophysical J; “FitzHugh-Nagumo Model” (2015) Encyclopedia of Comp Neuro
All-or-nothing response Post-inhibitory spike
![Page 34: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/34.jpg)
Two-dim. model: Hindmarsh-Rose model
( )( )
v f v u I
u g v u
= − +
= −
( ) ( )3 2 2,f v av bv g v c dv= − + = − +
Hindmarsh & Rose (1982) Nature; (1984) Proc R Soc Lond B
![Page 35: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/35.jpg)
Izhikevich model: quadratic and linear nullclines.
thresholdif 1, then and .v v v c u u d≥ = ← ← +
Quadratic v-nullcline and linear u-nullcline can describe both saddle-node and Andronov-Hopf bifurcations.
Figure 5.23, Dynamical Systems in Neuroscience
( )
2v v u Iu a bv u= − +
= −
![Page 36: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/36.jpg)
Izhikevich model reproduces various spiking patterns.
( )( )
20.04 5 140v v v u I t
u a bv u
= + + − +
= −
thresholdif 30, then and .v v v c u u d≥ = ← ← +
Izhikevich (2003) IEEE Trans Neural Networks
![Page 37: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/37.jpg)
Izhikevich model reproduces various spiking patterns.
Izhikevich (2003) IEEE Trans Neural Networks
![Page 38: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/38.jpg)
Adaptive-exponential model
( ) ( )
( )
T
T
V V
m L T L T
w L
dVC g V V g e w I tdtdw a V E wdt
τ
−∆= − − + ∆ − +
= − −
Brette & Gerstner (2005) J Neurophysiol
The adaptive-exponential model are popular to neurophysiologists because …
- It has a form similar to conventional two-dimensional models
- Its parameters are physiologically interpretable.
![Page 39: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/39.jpg)
What we left out: Neuron morphology (shape) does influence physiology (function)!
Mainen & Sejnowski (199) Nature
250 μm
100 ms
25 mV
![Page 40: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/40.jpg)
What we left out: Neuron morphology (shape) does influence physiology (function)!
Branco et al. (2010) Science
![Page 41: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/41.jpg)
Cable equation describes spike propagation.
“Cable Equation” (2015) Encyclopedia of Computational Neuroscience
![Page 42: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/42.jpg)
Rall model reduces to equivalent cylinder model.
“Equivalent Cylinder Model” (2015) Encyclopedia of Computational Neuroscience
With a set of assumptions about the morphological and electrical properties of dendrites, the complex branching structure of a dendritic tree can be reduced to a simple conductive cylinder.
2 23 3
1 22
30
GR d d
d
+=
If GR=1, then the cylinders 1 and 2 can be reduced to a single cylinder.
![Page 43: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/43.jpg)
Conclusions
- Neurons have a wide range of morphology (shapes) and physiology (functions).
- Many fundamental properties of subthreshold dynamics and spiking patterns can be captured by low-dimensional models.
- Models vary in their complexities: from a simple LIF model (just integrating and thresholding) to biophysically detailed Hodgkin-Huxley model.
- Phase-plane and bifurcation analyses are the powerful tool for understanding qualitative behaviors of a dynamical system without an explicit solution.
![Page 44: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/44.jpg)
Exercise
1. Read the following paper and derive a low-dimensional neuron model from a detailed HH-type model by linearizing around the resting potential.
Richardson et al. (2003) “From subthreshold to firing-rate resonance,” J Neurophysiol 89, 2538-2554.
2. Examine a qualitative behavior of the Izhikevich model by plotting a phase portrait:
a=0.02, b=0.2, c=-65, d=6, I=14 (constant).Then confirm your phase-plane analysis with the matlab code provided from Izhikevich’s site:http://www.izhikevich.org/publications/whichmod.htm#izhikevich
![Page 45: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/45.jpg)
Exercise
1. Simulate an integrate-and-fire model using the Euler method and evaluate how accurate the solution is. The Euler method is the simplest numerical integration method.
Brette, R., Rudolph, M., Carnevale, T., Hines, M., Beeman, D., Bower, J. M., ... & Zirpe, M. (2007). Simulation of networks of spiking neurons: a review of tools and strategies. Journal of computational neuroscience, 23(3), 349-398.
2. Simulate the Izhikevich model using standard parameters. Then plot the phase portraits in two dimensions.
( ) ( ) ( )( )m
v t RIv t v t
tt
τ− +
+∆
=∆+
![Page 46: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/46.jpg)
(0.02, 0.2, -65, 8) (0.02, 0.2, -55, 4) (0.02, 0.2, -50, 2) (0.1, 0.2, -65, 2)
(0.02, 0.25, -65, 0.05) (0.02, 0.2, -65, 0.05) (0.1, 0.26, -65, 8) (0.02, 0.25, -65, 2)
![Page 47: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/47.jpg)
% params for RS neuron:a = 0.02; b = 0.20; c = -65.; d = 8;
dt = 1/1000; % integration time step (s)T = 50.; % total simulation time (s)t0 = 0:dt:T; % time steps
%%v = zeros(length(t0),1); % voltage variableu = zeros(length(t0),1); % recovery variableI = zeros(length(t0),1); % input currentI(t0>1000/1000) = 150; spikes = zeros(length(t0),1); % spike timings
v(1) = -80; % initial voltageu(1) = 0; % initial recovery
for n=1:length(t0)-1v(n+1) = v(n) + dt*(0.04*v(n)^2+5*v(n)+140-u(n)+I(n));
% v(n+1) = v(n) + dt/2*(0.04*v(n)^2+5*v(n)+140-u(n)+I(n));% v(n+1) = v(n+1) + dt/2*(0.04*v(n+1)^2+5*v(n+1)+140-u(n)+I(n));
u(n+1) = u(n) + dt*a*(b*v(n)-u(n));
if v(n+1)>30v(n+1) = c;u(n+1) = u(n+1) + d;spikes(n+1) = 1;
end
end
figure(1); clf; subplot(211); plot(t0, v, 'k');subplot(212); plot(t0, u, 'k');
![Page 48: JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models](https://reader033.fdocuments.net/reader033/viewer/2022051318/58a784811a28abef478b5fb5/html5/thumbnails/48.jpg)
References
• Squire et al. (2008) “Fundamental Neuroscience,” Academic Press.
• Purves et al. (2004) “Neuroscience,” Sinauer Associates.
• Trappenberg (2010) “Fundamentals of Computational Neuroscience,” Oxford University Press, Chapters 2 & 3.
• Dayan & Abbott (2000) “Theoretical Neuroscience,” MIT Press, Chapter 5.
• Izhikevich (2007) “Dynamical Systems in Neuroscience,” MIT Press, Chapters 3 & 4.