Correlation-Induced Oscillations in Spatio-Temporal Excitable Systems

Andre LongtinPhysics Department, University of Ottawa

Ottawa, Canada

Co-Workers

Brent Doiron

Benjamin Lindner

Maurice Chacron Physics Department, University of Ottawa

Leonard MalerDepartment of Cellular and Molecular Medicine,

University of Ottawa

Joseph BastianDeparment of Zoology, University of Oklahoma

Synopsis

Introduction to weakly electric fish Oscillatory activity for communication but not Prey StimuliModeling I: Feedback is required Experimental verificationModeling 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 codingperipheral ↔ centralfeedforward ↔ feedbackin vivo ↔ in vitroStimuli: simple (sines etc…) ↔ naturalbehaviors: 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

Sensory Input

Higher Brain

Electroreceptor Neurons: Anatomy

Pore

SensoryEpithelium

Axon(To Higher Brain)

Biology: Weakly electric fish

90 100 110 120-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

mV

time (EOD cycles)

transepidermal voltage amplitude

90 95 100 105 110 115 120

0.7

0.8

0.9

1.0

1.1

1.2

1.3

mV

time (EOD cycles)

CIRCUITRY

Afferent Input

Higher Brain Areas

The ELL; first stage of sensory processing

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

Autocorrelation ISI Histogram

Local Stimuli (Dipole)

Fish

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

Global Stimuli

Autocorrelation ISI HistogramFish

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) .

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

),()( diii tVGtIV

dt

dV

Ii(t) – inputG(V,t-d) – interactions V

i

time

Pyramidal Cell Interactions – G(t-d)

))(exp()()(1

)(

0

2jm

N

j

tM

mjm tttt

N

gtG

j

The network is coupled through global delayed inhibitory feedback. The inhibitory response is modeled as a fast activating alpha function.

d

Pyramidal Cell Input - I(t)

Intrinsic Noise(biased Ornstein-Uhlenbeck process; =15 ms). Uncorrelated between neurons.

External Stimuli (Zero mean band passed Gaussian noise, 0-40Hz). Identical to experiments.

Pyramidal Cell

Ii(t) is composed of two types of “stimuli”

Network Model – Local Input

Autocorrelation Histogram

To mimic prey stimuli we apply the external stimulus to only one neuron

Network Model – Global Stimuli

To mimic communication stimuli we apply the external stimulus to all neurons equally.

Autocorrelation Histogram

Oscillation Mechanism

Local StimuliExternal Stimulus is applied heterogeneously across the network. No stimulus induced correlations.

Global StimuliExternal Stimulus is applied homogenously across the network. Significant stimulus induced 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 Autocorrelation

control

block

recover

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

)()(1)( tctct Gii

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.

Correlation Induced Oscillation

0 20 40 600.01

0.014

0.02

0.024

0.028 c=1

c=0.5

c=0

Frequency (Hz)

Pow

er

Linear Response

Consider the spike train from the ith neuron in our network, . Assuming weak inputs we have that the Fourier transform of the spike train is

)()( iji ttt

)()()(~)(~ ibgi XA

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().

(1)

Feedback Input

)(~)(~

)(~

)( jii dKN

gIX

Now consider the globally delay coupled LIF network used earlier. Let the “input” into neuron i be

Then it can be shown that for an infinite network we have

2

~2

22

)(~

1

)(~

)(~

Re2

)()()(

AKg

AKgAKg

AcSS

d

dd

bg

Fokker-Planck analysis on noisy Leaky Integrate-and-fire neurons

30

25

20

15

S (

spik

es2 /s

)

140120100806040200

frequency (Hz)

simulation c=1 theory c=1 simulations c=0 theory c=0

Shift in Coding Strategies

)()(

)()(

2

fPfP

fPfC

RRSS

SR

The spike train/stimulus coherence shifts from lowpass to highpass as we transition from local to global stimuli

Chacron, Doiron, Maler, Longtin, and Bastian,

Nature 423, 77-81 (May 1st 2003).

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).