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Correlation-Induced Oscillations in
Spatio-Temporal Excitable Systems
Andre Longtin
Physics Department, University of Ottawa
Ottawa, Canada
Co-Workers
Brent Doiron
Benjamin Lindner
Maurice Chacron
Physics Department, University of Ottawa
Leonard Maler
Department of Cellular and Molecular Medicine,
University of Ottawa
Joseph Bastian
Deparment of Zoology, University of Oklahoma
Synopsis
Introduction to weakly electric fish
Oscillatory activity for communication but not Prey Stimuli
Modeling I: Feedback is required
Experimental verification
Modeling II: stochastic oscillatory dynamics in a spatially extended
neural system
Doiron, Chacron, Maler, Longtin and Bastian, Nature 421 (Jan 30, 2003)
Weakly Electric Fish
Why study weakly electric fish?
(Biology)
from molecular to behavioral studies of neural
coding
peripheral ↔ central
feedforward ↔ feedback
in vivo ↔ in vitro
Stimuli: simple (sines etc…) ↔ natural
behaviors: simple ↔ evolved
(electrolocation) (electrocommunication)
Why study weakly electric fish?
(Mathematical Biology/Biophysics)
Single cell dynamics: simple ↔ complex
Linear, nonlinear, stochastic (get ready for noise!)
Information processing:
black box ↔ detailed biophysics
Math, Physics, Neuroscience, Computation
Applications: signal detection, novel circuitry,
prosthetic design (e.g. with feedback)
Sensory Neurons
ELL Pyramidal Cell
Higher
Brain
Sensory Input
Electroreceptor Neurons: Anatomy
Pore
Sensory
Epithelium
Axon
(To Higher Brain)
Biology: Weakly electric fish
1.3
1.5
1.2
1.0
0.5
1.0
mV
mV
1.1
transepidermal voltage
amplitude
0.0
0.9
-0.5
0.8
-1.0
0.7
-1.5
90
95
100
105
110
time (EOD cycles)
115
120
90
100
110
time (EOD cycles)
120
CIRCUITRY
The ELL; first stage
of sensory processing
Higher Brain
Areas
Afferent Input
Prey Stimuli
Electric fish prey on
small insects (water
fleas). These prey
excite only a fraction of
the electroreceptors that
line the fish’s skin. We
label this stimulation
geometry local
Communication Stimuli
Electric fish communicate
by modulating their own
electric field, giving a
specific input to other fish.
These communication calls
stimulate the entire surface
of the receiving fish’s skin.
We label this stimulation
geometry global
Pyramidal Cell Response to Prey-like
Stimuli
Fish
Local Stimuli (Dipole)
Autocorrelation
ISI Histogram
Random Amplitude Modulations
(RAMs) were applied locally to the skin
via a small dipole, mimicking prey
stimuli. The RAMs were a Gaussian
noise process (0-40Hz) .
Pyramidal Cell Response to
Communication-like Stimuli
Fish
Autocorrelation
ISI Histogram
RAMs were applied globally to the skin
via a large dipole, mimicking
communication stimuli. The RAMs were
generated by a Gaussian noise process
(0-40Hz) .
Global Stimuli
Model Pyramidal Cell
We model the ELL pyramidal cell network as an network of
Leaky Integrate and Fire (LIF) neurons. The membrane
potential of the ith neuron obeys the following dynamics
Ii(t) – input
G(V,t-d) – interactions
Vi
dVi
 Vi  I i (t )  G (V , t   d )
dt
time
Pyramidal Cell Interactions – G(t-d)
The network is coupled
through global delayed
inhibitory feedback. The
inhibitory response is modeled
as a fast activating alpha
function.
g
G (t ) 
N
N M j (t )

d
2

 (t  t jm ) exp( (t  t jm ))
j 1 m 0
Pyramidal Cell Input - I(t)
Ii(t) is composed of two
types of “stimuli”
Pyramidal Cell
Intrinsic Noise
External Stimuli (Zero mean band
(biased Ornstein-Uhlenbeck process;
=15 ms). Uncorrelated between
neurons.
passed Gaussian noise, 0-40Hz).
Identical to experiments.
Network Model – Local Input
Autocorrelation
To mimic prey stimuli we
apply the external stimulus
to only one neuron
Histogram
Network Model – Global Stimuli
Autocorrelation
To mimic communication
stimuli we apply the
external stimulus to all
neurons equally.
Histogram
Oscillation Mechanism
Global
Stimuli
Local
Stimuli
External
Stimulus
is is
External
Stimulus
applied
homogenously
applied
heterogeneously
across
the the
network.
across
network. No
Significant
stimulusstimulus
induced
induced
correlations.
correlations.
Correlated activity cause a
“wave” of inhibition after
a delay. This wave carves
out the oscillation.
Electrosensory Circuitry
The neural sensory
system of weakly
electric fish has a
well characterized
feedback pathway.
We applied a
sodium channel
blocker in order
to open the
feedback loop.
Experimental Verification
ISI Histogram
control
block
recover
Autocorrelation
Feedback: Open vs Closed Loop Architecture
Higher
Brain
Higher
Brain
Loop
time d
Correlated Stimuli in Experiments
Dipole 1
Dipole 2
Dipole 3
Dipole 4
The random signal emitted
from each dipole was
composed of an intrinsic, i(t),
and global source, G(t). The
relative strength of these two
sources was parameterized by
c, representing the covariance
between dipoles.
i (t )  1  ci (t )  cG (t )
Correlation Induced Oscillation
0.028
c=1
c=0.5
c=0
Power
0.024
0.02
0.014
0.01
0
20
40
Frequency (Hz)
60
Linear Response
Consider the spike train from the ith neuron in our network,
 i (t )    (t  tij ). Assuming weak inputs we have that
the Fourier transform of the spike train is
~i ( )  ~bg ( )  A( ) X i ( )
(1)
where A() is the susceptibility of the neuron determined
by the intrinsic properties of the cell. The Fourier transform
of the input (external + feedback) is given by Xi().
Feedback Input
Now consider the globally delay coupled LIF network used
earlier. Let the “input” into neuron i be
g ~
~
X i ( )   I i ( )  K  d ( ) ~ j ( )
N
Then it can be shown that for an infinite network we ~have

S ( )  Sbg ( )  c A( )
2
2
)
~
~
2 Re gK d A( )  g K  d A( )
~
1  g K  d A( )
2
2
Fokker-Planck analysis on noisy
Leaky Integrate-and-fire neurons
25
2
S (spikes /s)
30
20
simulation c=1
theory c=1
simulations c=0
theory c=0
15
0
20
40
60
80
100
frequency (Hz)
120
140
Shift in Coding Strategies
The spike train/stimulus
coherence shifts from
lowpass to highpass as
we transition from local
to global stimuli
C( f ) 
Chacron, Doiron, Maler, Longtin, and Bastian,
Nature 423, 77-81 (May 1st 2003).
PSR ( f )
2
PSS ( f ) PRR ( f )
Spike Time Reliability
When high-frequency stimuli (40-60 Hz) are given, spike time
reliability is increased dramatically when the stimulus is applied
globally.
Conclusion
 Electric fish use delayed inhibitory feedback to
differentially respond to communication vs. prey
stimuli.
 Our work shows how a sensory system adapts its
processing to its environment (local vs. global).