Transcript "TOPICS IN THEORETICAL NEUROBIOLOGY"

STOCHASTIC MODELS IN
NEUROSCIENCE
MARSEILLE, FRANCE
JANUARY 2010
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• THANKS TO MICHELE, NILS AND
SIMONA FOR ORGANIZING SUCH
AN INTERESTING AND ENJOYABLE
MEETING.
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THE EFFECTS OF WEAK
NOISE ON RHYTHMIC
NEURONAL ACTIVITY
HENRY C. TUCKWELL
MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE
SCIENCES, INSELSTRASSE 22-26, LEIPZIG, GERMANY
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PLAN OF TALK
WEAK NOISE EFFECTS ARE CONSIDERED IN SINGLE NEURON MODELS
A.
THE HODGKIN-HUXLEY SYSTEM OF ODE’S (POINT MODEL)
B.
HH AS PACEMAKER (ODE)
C.
A MULTI-COMPONENT BIOLOGICAL PACEMAKER MODEL
D.
THE HODGKIN-HUXLEY SYSTEM OF PDE’S (CABLE MODEL)
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A: HH ODE MODEL WITH NOISE
BACKGROUND
• We had been considering the effects of noise on
coupled type 1 (QIF) neurons ( See Gutkin, Jost &
Tuckwell, Theory in Biosciences 127, 135-139 (2008)
& Europhysics Letters 81, 20005 (2008) )
and commenced a similar study of coupled HH
neurons.
However, finding that minima arose in the activity,
at various values of the coupling strength, and
especially zero coupling, led us to consider single
HH neurons with noise.
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Introduction
• Recall that noise may induce firing. e.g. a leaky integrate and
fire model (but not an integrate and fire model) neuron with
periodic impulsive excitation at frequency f with amplitude a>0
may never fire an action potential, but if the input is Poisson
with the same mean frequency f>0 , the neuron will fire in a
finite time with probability 1 – see Introduction to Theoret
Neurobiol Ch 3 and later chapters.
• Also, noise even of zero mean increases the mean membrane
potential in the standard LIF model.
• However, here we are mainly concerned with the inhibitory
effects of noise on neuronal spiking.
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For the HH ODE model with additive
noise
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•
USING STANDARD PARAMETER VALUES THE CRITICAL VALUE OF µ TO INDUCE
REPETITIVE FIRING (HOPF BIFURCATION) IS ABOUT 6.44. FOR VARIOUS VALUES OF µ
AND σ ONE OBTAINS RESULTS SUCH AS THESE (µ=6.6)
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WITH CORRESPONDING ORBITS
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•
WE INVESTIGATED THE DEPENDENCE OF THE NUMBER OF SPIKES
OVER A LIMITED TIME PERIOD ON THE NOISE AMPLITUDE FOR
VARIOUS VALUES OF THE MEAN CURRENT
AND OBTAINED THE FOLLOWING IN THE ADDITIVE NOISE
CASE.
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HH, ADDITIVE WHITE NOISE
SPIKES OVER 1000 MSEC
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FOR THE CONDUCTANCE-BASED INPUT WE HAVE
AN INPUT CURRENT OF THE FORM
I_c=g_E(V_E-V) + g_I(V_I-V)
where V_E, V_I are the excitatory and inhibitory reversal
potentials
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with the following results:
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WE ALSO CONSIDERED SWITCHING THE NOISE ON AT A
RANDOM TIME AFTER REPETITIVE SPIKING WAS ESTABLISHED
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In addition
we
considered
choosing
initial
conditions
randomly
with result as
shown here
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THE APPROXIMATE BASINS OF ATTRACTION OF THE REST
POINT AND LIMIT CYCLE (WHICH DEPEND ON MU)
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CONSIDER HOW THE PROCESS LEAVES THE LIMIT
CYCLE
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WE COLLECTED DATA ON WHERE THE EXIT POINTS WERE
ROUGHLY LOCATED, WHICH CAN BE COMPARED WITH THE
ESTIMATE OF THE BASINS OF ATTRACTION
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WHEN THE NOISE IS “LARGE” THE PROCESS MAY QUITE
FREQUENTLY MAKE TRANSITIONS TO AND FROM THE BASINS
OF ATTRACTION OF THE STABLE POINT AND THE LIMIT CYCLE
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WITH INTERSPIKE INTERVAL HISTOGRAMS AS
FOLLOWS
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• One sees that there is a “competition”
between the tendency of noise to stop
the spiking and the tendency for it to
induce spiking.
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MARKOV THEORY
WE CONSIDER THE NATURE OF THE ATTRACTORS
OF WHICH, FOR MEAN CURRENTS NOT MUCH
GREATER THAN THE CRITICAL VALUE, THERE
ARE TWO : A STABLE REST STATE AND A
STABLE LIMIT CYCLE.
JUST PAST THE CRITICAL VALUE THE BASIN OF
ATTRACTION (BOA) OF THE LIMIT CYCLE IS
SMALL AND A SMALL NOISY SIGNAL (OR ANY)
CAN KICK THE DYNAMICS INTO THE BOA OF
THE STABLE REST POINT – THUS
TERMINATING THE SPIKING.
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Theory: Exit-time theory for Markov processes
• Theorem: The process switches from spiking to nonspiking states (and vice-versa) in a finite time with
probability one. The expected times which the
system remains in one or the other state are the
solutions of linear partial differential equations given
below
• Sketch proof
• The process (V,m,h,n) has an infinitesimal
operator L. That is, the transition density p
satisfies a Kolmogorov equation
•
∂p/ ∂ t = Lp
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• The prob pL of leaving the BOA BL of the limit cycle satisfies
•
LpL =0 on BL (*)
• with boundary condition
• pL =1.
• The solution of * is pL = a constant. Hence, because process is
continuous, pL =1 throughout BL.
• Similarly for the prob pR of leaving the BOA BR of the rest state.
Standard theory gives that the expected time to stay in the
spiking state
satisfies LFL =-1 on BL with boundary condition FL = 0.
Similarly for the expected time to leave BR. The behaviour of
the system is thus characterized by a sequence of alternate exit
times from BL and BR.
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MOMENT ANALYSIS
We have also sought explanations of
these phenomena “analytically” . Thus we
have found the moment equations for an
HH neuron with noise – in the additive
noise case there are 14 de’s.
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FOR HH THE COMPONENTS ARE,
USING STANDARD SYMBOLS, X1=V,
X2=n, X3=m AND X4=h.
• The means are denoted by m_1, m_2, m_3, m_4
and there are 4 variances C_11, C_22, C_33 and C_44
together with another
6 covariances C_12, C_13, C_14, C_23, C_24, C_34 =
14 1st and 2nd order moments.
For example, C_24 = Cov (n(t), h(t))
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FAIRLY LENGTHY CALCULATIONS GIVE FOR EXAMPLE, THE DE’S
FOR THE MEAN AND VARIANCE OF THE ,VOLTAGE VARIABLE
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• OTHER EQUATIONS INVOLVE THE ALPHA’S AND BETA’S
AND THEIR DERIVS E.G.
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FOLLOWING SHOW THE MEAN AND VARIANCE OF THE
VOLTAGE: THE FIRST TWO SETS OF RESULTS ARE FOR SMALL
NOISE AND SHOW THE EXCELLENT AGREEMENT BETWEEN
ANALYTICAL AND SIMULATION RESULTS
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THIS SHOWS HOW A SMALL NOISE MAY SOON GIVE
RISE TO A LARGE VARIANCE AND POSSIBLY DRIVE
THE SYSTEM TO REST
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THIS IS FURTHER ILLUSTRATED BY THIS SAMPLE PATH
PICTURE. WE HAVE ALSO DETERMINED THAT THE SPEED OF
THE PROCESS IS SMALL NEAR THE EXIT POINT. THE
REDUCTION IN VELOCITY CONTRIBUTES TO THE EXIT.
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EXPERIMENTAL CONFIRMATION OF THE SILENCING OF NEURONAL
ACTIVITY BY NOISE CAME IN 2006 ON SQUID AXON – AN ARTICLE BY
Paydarfar, Forger & Clay: Noisy inputs and the induction of on-off
switching behavior in a neuronal pacemaker. J. Neurophysiol. 96, 33383348. 8 AXONS WERE EXAMINED.
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B. HH AS “PACEMAKER”
Writing the HH auxiliary equations in the following
form
with activation and inactivation steady state values
and time constants as a function of voltage, one
may turn the HH neuron into a spontaneously
firing “cell” by shifting, for example, the half
activation potential to -30.5 mV from about -28.4
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mV (assumed resting at -55 mV).
Normally the HH neuron does not fire with zero input or with
hyperpolarizing input current, but with V_1/2 = -30.5 there is a threshold
for repetitive spiking around +1.8 nA (standard model). See below.
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Now with an initial value V(0)=V_R + 10, periodic spike trains were
impeded with noise with a similar behaviour as in the standard HH
case.
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C. BIOLOGICAL PACEMAKER WITH
NOISE
•
•
•
•
• BACKGROUND
Interest here focuses on whether pacemaker activity,
which often performs vital biological functions,
could be stopped by a small noise.
I had been studying the brain circuitry involved in
stress, which involves interactions between the
nervous and endocrine systems.
The structures involved are many and diverse and
only a partial picture is able to be given.
I have put together as a first attempt a map of the
brain components involved.
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MODELS
• Many pacemaker or pacemaker-like models have been proposed
over the last 18 years including
• McCormick and Huguenard 1992 Thalamic relay
• Schild et al. 1993 Nucleus tractatus solitarii (brainstem)
• Destexhe et al. 1994 Thalamic reticular nucleus
• Rybak et al. 1997 Respiratory neurons……..
• up to more recently…..
• Rhodes and Llinas 2005 Thalamic relay
• Putzier et al 2009 Dopamine neurons in SN (empirical with virtual
• L-type channels)
• See also “Thalamocortical Assemblies”, Destexhe & Sejnowski 2002
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The are 10 currents included in the present model. Except for
I_K(Ca), these channels are all voltage-dependent in the present
model.
• In addition the internal calcium concentration
Ca_i is included in the model.
• This is determined by some calcium currents
with allowances for buffering and pumping.
• The activation and inactivation curves are
taken from various references. The 10
maximal conductances are estimated from
area measurements and channel densities. In
total about 75 parameters must be specified.
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Model with noise
+ steady current (which could be synaptic in origin)
• CdV/dt = -Σ_i g_i (V-V_i) +
-mu + σw
where mu is a constant, g_i is
conductance, w is GWN and the sum is
over the above 10 channel-types. V is
membrane potential, not depolarization.
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Rhythmic firing near threshold10-component process
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Ten trials small noise
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Ten trials larger noise
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Ten trials with sigma = 0.6
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D: THE HODGKIN-HUXLEY PDES WITH
NOISE
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RESULTS 1 : no noise: X_1=0.2, L=6
VARIOUS MU. CURRENT LEFT ON.
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RESULTS 2: NUMBER OF SPIKES ON (0,L) UP TO T=160 FOR
TWO VALUES OF X_1: BIFURCATION TO REPETITIVE FIRING AT
ABOUT MU=6
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Results 3: Inhibitory effects of noise applied throughout whole
cylinder: Signal to x_1=0.1. E[N]=expected number of spikes on
(0,L) at t=160 ms. 50 trials..
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RESULTS 4: A TRIAL WITH LARGER NOISE SHOWING
HOW SECONDARY SPIKES MAY LEAD TO AN END
RESULT WITH NO SPIKE ON (0,L)
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RESULTS 5 :NOISE ON SMALL INTERVALS
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Comments
• The effects of the spatial distribution of noise were
surprising. The orbits at each space point are about the
same for the travelling wave, but if there is no “signal”
then the noise has no effect
• Since the 70’s travelling waves have been much studied
in deterministic reaction-diffusion systems but there are
hardly any results for stochastic such systems.
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CONCLUSIONS
•
Weak noise in the HH system of ODE’s may inhibit rhythmic spiking for
very long time periods near the critical value mu_c
•
Near mu_c, with increasing noise, a pronounced minimum occurs in the
expected number of spikes which is sometimes called “inverse
stochastic resonance”
•
Large noise causes transitions back and forth from spiking to rest with
a frequency that actually increases the mean spike rate relative to no
noise
•
Weak noise may stop the activity of biological pacemakers. “Inverse
stochastic resonance” may occur.
•
In the HH system of PDE’s, weak noise may effectively stop rhythmic
firing near the critical value, but the spatial distribution of the noise is
important.
•
Collaborators: Juergen Jost MIS (HH ODE’s and PDE’s), Boris Gutkin
ENS (HH ODE’s). See Phys Rev E (2009), Physica A (2009),
Naturwissenschaften (2009) and submitted (2010).
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