Neurodynamics

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Neurodynamics The Nonlinear Dynamics of Neurons
and Networks H
enry D. I. Abarbanel hdia_at_jacobi.ucsd.edu Departme
nt of Physics, and Marine Physical Laboratory
(Scripps Institution of Oceanography) and Misha
I. Rabinovich rabin_at_landau.ucsd.edu Institute for
Nonlinear Science University of California, San
Diego La Jolla, CA 93093-0402 USA
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This set of lectures is concerned with the
dynamical understanding of neurons as nonlinear
oscillators, of their synapses as dynamical
connections, and of dynamical networks (usually
small) of these components whose function we seek
to understand. Understanding such networks
and components means to us modeling their
electrical activity and matching various
nonlinear statistical characteristics of
experiments and models.
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Using this understanding we will discuss how we
seek to reproduce the functional action of the
networks in analog electrical circuitry. A
physicists viewpoint. How we test that
circuitry against the behavior of biological
neurons is also an essential question. We answer
it in detail using experiments on electrical
neurons, identical experiments on biological
neurons, and hybrid experiments.
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The contents of these lectures have been
worked out in collaboration with Matt Kennel,
Misha Sushchik, Ramon Huerta, Rob Elson, Allen
Selverston, Pablo Varona, Lev Tsimring, Attila
Szücs, Martin Falcke, Nikolai Rulkov, Sasha
Volkovskii, Manuel Eguia, Evren Tumer, Reynaldo
Pinto, and others Work done at UCSDs Institute
for Nonlinear Science
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• Outline of Topics
• Dynamics of Individual Neurons
• Experimental Observations
• Hodgkin-Huxley Models
• Tools of Nonlinear Dynamics for Analysis of Time
  Series application to membrane voltage
  measurements
• Analysis of a Nonlinear Electrical Circuit
• Some other examples
• Analysis of Bursting/Spiking Neurons from a
  Central Pattern Generator

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• Outline of Topics
• Framework for Reduced Models Building and
  testing reduced models
• Hybrid Circuits
• Synchronization of Neurons
• Information Theory
• Information Theory in Neuroscience
• The Neural Code
• Reading the Neural Code

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Schematic diagram of a section of the lipid
bilayer that forms the cell membrane showing two
ion channels embedded in it. The membrane is 3 to
4 nm thick and the ion channels are about 10 nm
long. (Adapted from Hille, 1992.)
Neurons are cells with proteins puncturing the
cell membrane which allow, in a voltage dependent
manner, the passage of ions into and from the
cellular interior.
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A) A cortical pyramidal cell. These are the
primary excitatory neurons of the cerebral
cortex. Pyramidal cell axons branch locally,
sending axon collaterals to synapse with nearby
neurons, and also project more distally to
conduct signals to other parts of the brain and
nervous system. B) A Purkinje cell of the
cerebellum. Purkinje cell axons transmit the
output of the cerebellar cortex. C) A stellate
cell of the cerebral cortex. Stellate cells are
one of a large class of cells that provide
inhibitory input to the neurons of the cerebral
cortex. To give an idea of scale, these figures
are magnified about 150 fold. (Drawings from
Cajal, 1911 figure from Dowling, 1992.) From
Dayan and Abbott, 2000.
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Orders of Magnitude Transport must exceed thermal
fluctuations.
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The capacitance and membrane resistance of a
neuron considered as a single compartment. The
membrane resistance Rm determines the size of the
membrane potential deviation V caused by a small
current Ie entering through an electrode. Cm and
Rm, are related to the specific membrane
capacitance and resistance, cm and rm, as shown.
rm may vary considerably under different
conditions and for different neurons. For Vm
-70mV, about 109 singly charged ions are
transported across the membrane. This is the
charge transported by 0.7 nA in 100ms. Dayan and
Abbott, 2000
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Recording of the current passing through a
single ion channel. This is a synaptic receptor
channel sensitive to the neurotransmitter
acetylcholine. A small amount of acetylcholine
was applied to the preparation to produce
occasional channel openings. In the open state,
the channel passes 6.6 pA at a holding potential
of -140 mV. This is equivalent to more than 107
charges per second passing through the channel
and corresponds to an open channel conductance of
47 pS. (From Hille, 1992.)
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There are many ion species playing a role in
neural activity. Na is an important one.
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As membrane voltage rises from its equilibrium
value ion channels in the membrane open.
flows in, and potential tends to rise,
flows out and potential tends to fall. This
pulse of current gives rise to a voltage
pulsean action potential. Other ions (
) also play a role.
So we need a model of the channel currents
Hodgkin-Huxley circa 1952 contributions by Katz
as well Hille (1992).
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Empirical
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As membrane voltage increases above a threshold
contained in, , open(t)
rises and the current grows. Then close(t)
decreases, effectively turning off the current.
This yields a spike or action potential. The
competition between incoming and outgoing
currents causes oscillations or bursts of action
potentials.
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Aplysia a widely studied mollusc is a favorite
neurophysiological preparation. It has big
active, but slow, neural spiking patterns.
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Courtesy of Carmen Canavier, University of New
Orleans, Louisiana, we have the following
Hodgkin-Huxley model for the R15 neuron of
Aplysia
and so forth for all of the opening and closing
variables hj(t). This model neuron also has Ca2
uptake and release dynamics.
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Since the output of this model neuron (and the
real R15 neuron as well) is a train of more or
less identical spikes or action potentials, any
information carried by modulating this neuron
must be in the ISIs
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This behavior is quite typical of neural models.
For large positive injected DC currents, we see
periodic spiking. Biologists call this tonic
spiking it is a stable limit cycle in the system
phase space. As the DC current is decreased,
this limit cycle becomes unstable and the neural
oscillations go through a sequence of
bifurcations including chaotic states.
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Another model neuron well see this one later
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Now we turn to the analysis of voltage (or
other) signals from neurons. Typically an
electrode of size about one micron or less is
inserted through the cell membrane. We use drawn
glass tubes with 3M KCl as a conductor. The tip
resistance is order 10 M Measurements are made
fast enough to sample spikes quite well. This
means sampling rates fs of 2-10 kHz. The result
is a time series
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Analyzing Signals from Nonlinear Sources
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Comparison between linear and nonlinear signal
analysis tasks Finding the right state
space--- Time Delay Reconstruction from One
Observed Variable Average Mutual
Information False Nearest NeighborsGlobal and
Local Characterizing the Attractor Dimensions,
Lyapunov Exponents, UPOs
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Predicting Control to UPOs Predicting A from
B Determing one (hard) measurement from another
(easy) measurement Synchronization Input/Output
SystemsReading the Neural Code
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This is the wrong way to look at the signal
V(t). It contains three degrees of freedom which
are irregular, actually nonperiodic, in time.
Fourier spectra accurately pick out periodicities
and associate them with the regular, linear
resonse of a systemhere an electronic
circuit. Instead we look for three independent
views of the dynamics of this circuit, and we
plot the vectors V(t),V(t-T),V(t-2T).
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Finding the State Space in which to
work Nonlinear systems with complex behavior
must have two or more dynamical variables for
discrete time or three or more for continuous
timedifferential equations We observe a single
time series and we want to construct the space
from this.
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For finite sampling time, ?s, these are high
pass filters. They emphasize errors rather than
the careful measurements s(t). In ds(t )/dt
the only new information is s(n1) in the second
derivative, the only new information is s(n2).
It was the idea of David Ruelle to use just these
pieces of information as coordinates in a proxy
state space.
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T is an integer in the second line the sampling
time is taken as unity.
s(n-T) contains information about all other
(unobserved) dynamical variables of the system
because they act during time T and affect the
changes in s(n) as it evolves from s(n-T). If
the signal comes from a finite dimensional
system, then some value of d should capture the
full dynamics. We need T large enough so the
dynamics at s(n-T) affects s(n) but not so large
that the natural instabilities of nonlinear
systems cause s(n-T) and s(n) to be numerically
independent.
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We start with the example of s(t) sin(2?ft)
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Two questions now How are we to choose T so
s(n) and s(nT) are two independent looks at the
dynamics of the system? How are we to choose d
so we have enough components of the proxy data
vector y(t)?
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To select T so s(n) and s(n-T) are independent,
we need a nonlinear measure of correlation
between them. We choose to use the average
mutual information between measurements T
apartaveraged over the time series.
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Role of information in nonlinear dynamics
Entropy, is
proportional to
TIME
XX
X X
Linear
Resolution in State Space
X
X
Nonlinear
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Short digression on information theory This
describes the information carried by an observed
variable which takes distributed values. The
variable ak is measured at times k
-2,-1,0,1,2, in units of some sampling time.
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This answers the question how much, on the
average over all measurements of A and B
quantities do we learn about the possible
outcome a of an A quantity when we measure a B
quantity and observe b? Again, if the logarithm
is taken to base 2, the units of these
information quantities is bits.