Stochastic Models of Neurons [618547]

Stochastic Models of Neurons
Irina-Floriana Stanciu
Stochastic Models of Neurons
Irina-Floriana Stanciu
Objectives
1. Introduction. What is a neuron?
2. Hodgkin-Huxley Model
3. Fitzhugh-Nagumo Model
4. Processes & Results
5. Conclusion
Introduction
Neurons are dened as the basic cells of the nervous system. They
have a cell body (soma), dendrites and an axon. Organised in
neural circuits, these cells communicate information all over the
body, receive stimuli from the external world and send commands
to our muscles. They interact with other neurons through action
potential, using electrical signals called synapses. Neurons receive
electrical signals through dendrites. Dendrite comes from the
Greek word dendron which means tree, therefore they may look
similar to the branches of an oak tree. In 1886, Camillo Golgi
suggested that their function is to collect nutrients for the neuron.
The truth is that they collect information in the form of synaptic
input.
Hodgkin-Huxley Model
The Hodgkin-Huxley model is based on a simple electrical cir-
cuit with batteries, resistors and capacitors. The current is rep-
resented by ions which pass through the membrane (considered
resistors) or charge capacitors. It is formed of three ionic currents:
sodium current, potassium current and leakage current that con-
sists mainly of Cl ions (Hodgkin and Huxley 1952).
Process & Results
The Hodgkin-Huxley model was developed using a set of four nonlinear dierential equations that
approximate the electrical characteristics of excitable cells (Nelson 2004).
Cm
dV
dt=IGKn4(VEKGNam3h(VENa)GL(VEL) (1)
dn
dt=n(V)(1n)n(V)n (2)
dm
dt=m(V)(1m)m(V)m (3)
dh
dt=h(V)(1h)h(V)h (4)
Them,n,hvariables are in the range [0 ;1] and follow the dierential equations: mandh
represent the activation and inactivation of the sodium current and nrepresents the activation of
the potassium current.
The plots represent the Stimulus Graph , the Neuron Potential with Two Spikes Graph and
theLimit Cycles Graph . Using the function odeint in Python it was observed that during
50 miliseconds ( minimumtime = 0:0 andmaximumtime = 50:0) we have: GK= 36:0,
GNa= 120:0,GL= 0:3:(average potassium, sodium and leak channel conductance per unit
area (mS=cm2)),Cm= 1:0 (membrane capacitance per unit area ( F=cm2),VK=12:0
(potassium potential (mV)), VNa= 115:0 (sodium potential), VL= 10:613 (leak potential), their
conductances specic forms are:
Potassium: GK=G0
Kn4;G0
K=constant .and Sodium:GNa=G0
Nam3h;G0
Na=constant:
Using Kircho's Law ,Pn
k=1Ik= 0 and Ohm's Law I=V
RthenItotal =INa+IK+IL+Iext
withIx=Gx(ExVm) for each current.
In order to get the plots, the rates function of the potassium ion-channel rate and the sodium
ion-channel were analysed, the potential values were introduced, the way stimulus works was
evaluated, the derivatives were computed and the ODEs system solved. This process conducted
to the HH model formula.
Fitzhugh-Nagumo Model
Fitzhugh-Nagumo model represents a simplied two-dimensional circuit based on the HH model.
In 1961, Fitzhugh observed that the variables nandhhave slower kinetics than mand the
following year Nagumo built the equivalent circuit which contains the Van Der Pol oscillator,
the reason why it was previously named the Bonhoeer-van der Pol oscillator .
Process & Results
Fitzhugh and Nagumo derived a two dimensional model for an excitable neuron. The model is char-
acterised by two nonlinear ordinary dierential equations one describing the evolution of the neuronal
membrane voltage and the other one the slower action of the sodium and potassium channels deactiva-
tion. The used values are a=0.7, b=0.8, = 13:
The system is stable forjaj>1 and unstable forjaj<1. Near the bifurcation jaj= 1, the model is a
nonlinear dynamical system expressing excitability. It will have large pulses when jajis a bit larger than
one. Small-amplitude trajectories are expressed by small deviations and they represent the subthreshold
responses. Large-amplitude trajectories will appear if the deviations are larger than the threshold. The
suprathreshold response corresponds to ring a spike. The White Gaussian noise expresses the eects
of the random processes found in nature.
dx
dt=A(x;y) +I+(t) (5)
dy
dt=B(x;y) (6)
A(x;y) =ax3+bx2+cx+hy (7)
B(x;y) =ex+fy+g (8)
Using the odeint andtznag functions in Python, the white noise plot was represented.
It is called Gaussian correlated process because is has the correlation function: k() =N(),
N=constant. The power spectrum is:
S[x;] =NR1
1()ei
d=N
The noise term is x(t) satises both E[x(t)] = 0 and k() =Efx(t)x(t+)g=()fx(t)gwith
t[0;T] represents the standard white noise when N= 1.
Conclusion
The Hodgkin-Huxley model is fundamental in order to understand the connection between neurons
because it can predict action potentials that express a lot of information about their signals. The
Fitzhugh-Nagumo model is a simplied model of Hodgkin-Huxley and it is used to analyse dierent
neural ring patterns.
References
1. Cannon, Robert C., Cian O'Donnell, Matthew F. Nolan, and Lyle J. Graham. Stochastic Ion
Channel Gating in Dendritic Neurons: Morphology Dependence and Probabilistic Synaptic
Activation of Dendritic Spikes (Stochastic Ion Channels and Neuronal Morphology) .PLoS
Computational Biology 6.8 (2010): E1000886.
2. Bresslo, Paul C, Stuart Antman, Leslie Greengard, Philip Holmes, Leon Glass, Robert Kohn,
P. S Krishnaprasad, James D Murray, and Shankar Sastry Stochastic Processes in Cell Biology
Cham: Springer International, 2014. Interdisciplinary Applied Mathematics.
3. Hausser, Michael. (2000). The Hodgkin-Huxley theory of the action potential. Nature neuro-
science. 3 Suppl. 1165. 10.1038/81426.
4. Faghih, Rose Savla, Ketan Dahleh, Munther N Brown, Emery. (2010). The FitzHugh-Nagumo
Model: Firing Modes with Time-varying Parameters&Parameter Estimation.Conf.2010.4116-
9.10.1109/IEMBS.2010.5627326.
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