Transcript 72x48 Poster Template - Duke University Mathematics Department

Phase Delayed Inhibition in the Rat Barrel Cortex
Runjing (Bryan) Liu; Dr. Mainak Patel
Department of Mathematics, Duke University
Abstract
Phase delayed inhibition is a mechanism that enables the detection of
synchrony in a group of neurons. This mechanism is found in many parts of
the brain such as the neocortex, the hippocampus, the cerebellum, and the
amygdala (Bruno, 2011). The project I worked on dealt specifically with the
rat barrel cortex, the region of the somatosensory cortex that corresponds
to whisker deflections. Whisker deflections first stimulate a specific group
of neurons in the thalamus which send input to the barrel cortex. The barrel
cortex is then able to detect synchrony levels in the thalamic neurons using
phase delayed inhibition. Synchrony levels in the thalamus encodes
whisker deflection velocity, and this provides the rat information about its
surrounding environment (Bruno, 2011). To study phase delayed inhibition
mathematically, I used the Wilson-Cowan neuron population model and the
integrate-and-fire neuron model. Using the Wilson-Cowan model, I
examined how a generic phase delayed inhibition network acted as a
synchrony detector. Then, I applied the integrate-and-fire model to phase
delayed inhibition in the rat barrel cortex and proposed a mechanism for
sensory adaptation: that sensory adaptation is a result of decreased
synaptic strengths.
Introduction
Synchrony is a common way to encode information in the brain; for
example, whisker deflection velocity is encoded by thalamic neurons firing
in synchrony (Bruno, 2011). Phase delayed inhibition is one way to decode
this synchrony. The network looks as follows in figure 1:
•Notice that for every
encoder neuron that is
firing, the decoder first
receives an excitatory
signal followed by an
inhibitory signal.
•Thus, if the population
of encoder neurons are
firing asynchronously,
the decoder will receive
a mix of excitatory and
inhibitory inputs, and it
Figure 1: The basic phase delayed inhibition network
cannot fire.
•However, if the encoder neurons are firing synchronously, the decoder
neuron will receive a large block of excitatory input before receiving a large
block of the inhibitory input; this results in a window of time during which
the decoder neuron can fire.
Mathematical Model: Wilson-Cowan
The model developed by Wilson and Cowan is comprised of two
differential equations, one modeling the population of excitatory neurons,
the other modeling the population of inhibitory neurons (Wilson and
Cowan, 1972).
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Results
Tailoring the model to fit the phase delayed inhibition network, we obtain:
where E is the proportion of the excitatory neurons that are firing at a given
time, and I is the proportion of the inhibitory neurons that are firing at a
given time. c, τi and τe are constants, and P(t) is the stimulus received by
the encoder neuron population. δi and δe are sigmoid functions of the form:
δi and δe give the proportion of inhibitory and excitatory neurons that are
firing as a function of input.
Finally, the decoder neuron activity was modeled simply by:
The -2I(t) signifies that the inhibitory signal received by the decoder is twice
as strong as the excitatory signal.
A synchrony detector
Using the Wilson-Cowan model, I tested whether or not phase delayed
inhibition could truly act as a synchrony detector. To model synchrony, I
based my P(t) on a periodic Gaussian distribution with mean µ being the
middle
. of the period, and standard deviation σ=ω(1-s). s is a measure of
synchrony ranging from 0 (least synchronous) to 1 (most synchronous).
(See figure 2).
Figure 2: P(t) at three different levels of
synchrony
Figure 3 the effect of synchrony on decoder
activity
Insert plot here
Figure 5: A phase plane and the nullclines at a
fixed time t
Figure 6: trajectories at two different synchrony
values
In analyzing the trajectory as time passes, I find that:
•The trajectory is trying to follow the moving fixed point as time elapses
• How well the trajectory can follow the fixed point is determined by Ti and
Te in equations 1 and 2). In my model, Ti is set larger than Te (Ti=5, Te=2),
so the trajectory follows the fixed point better in the E direction
•At high synchronies, the movement of the fixed point is very rapid, and the
trajectory will initially be flat: due to the larger Ti, the trajectory closely
follows the fixed point in the E direction but not the I direction (figure 6, red
curve).
•At low synchronies the fixed point is moving slower. There is not this flat
portion because the trajectory can keep up better with this slower moving
fixed point (figure 6, green curve).
•It is during this flat portion that the decoder can fire because here E >>I
Rat Barrel Cortex: the physiology
One particular system that uses phase delayed inhibition to decode
synchrony is the rat barrel cortex, which corresponds to the detection of
whisker movement. Whisker movements first stimulate thalamic neurons;
these thalamic neurons may fire
synchronously or asynchronously
depending on the velocity of the
whisker deflection; higher whisker
deflection velocities leads to higher
values of synchronies, and this
synchrony is then decoded by cells
in the barrel cortex using phase
delayed inhibition (Bruno, 2011). Figure 7: phase delayed inhibition in the barrel cortex
Sensory Adaptation
Figure 4: Phase delayed inhibition under two different synchrony levels. Note how the decoder fires
periodically at high synchrony (right), but is unable to fire at the lower synchrony (left)
From the figures 3 and 4, we see that phase delayed inhibition can in fact
act as a synchrony filter. The decoder cannot fire at low synchronies, but it
can fire at higher synchronies.
Phase Plane Analysis
Analyzing equations 1 and 2 in the phase plane (figure 5), I found that the
fixed point:
•was stable at all points in time
•simply goes up and down the blue line in figure 6 as time elapses
•I also assumed that it was globally attracting
I tried several ways to explain how these phenomenon can occur. One way
to explain the decrease in thalamic to inhibitory signal was to decrease the
synaptic strength between each thalamic and inhibitory neuron. By
decreasing synaptic strength, I noticed:
•the delay between the inhibitory and the thalamic neurons increased
(see figure 8, left).
• the synchrony of the inhibitory neuron increases (or standard deviation
decreases), (see figure 8, right). Increasing the synchrony of the inhibitory
neurons is important because this balances out the effect of decreasing
thalamic synchrony, which decreases the synchrony of the inhibitory
neurons. Therefore, by simultaneously decreasing the synaptic strength
(making the inhibitory population more synchronous) and lowering thalamic
synchrony (making the inhibitory population less synchronous), we have a
balance in which the synchrony of the inhibitory neurons should not
change.
Going forward, I used the integrate-and-fire neuron model to study a
phenomenon in the rat barrel cortex known as sensory adaptation. This
phenomenon occurs when a whisker is repetitively deflected, and it results
in the decoder neuron becoming less responsive to whisker deflection. In
this project, I was more interested in the excitatory (thalamic) and inhibitory
(FS) cell interaction after adaptation. Adaptation causes these three
phenomenon as observed by Gabernet et al. (2005):
1. Decreased thalamic to inhibitory signal
2. Decreased synchrony of thalamic cells, but constant synchrony for
inhibitory cells
3. Increased latency between thalamic cells firing and inhibitory cells firing
Figure 8: effects of changing synaptic strengths
Summary
1.
2.
3.
Phase delayed inhibition can in fact act as a synchrony detector: the
decoder neuron only fires when synchrony is above a certain value.
The time constants Ti and Te played an important role in determining
how phase delayed inhibition works; we discussed how it was having
Ti>Te that allowed phase delayed inhibition to be a synchrony
detector. Physiologically, this means that the inhibitory neurons must
respond slower than excitatory neurons.
Simultaneously decreasing synaptic strength and thalamic synchrony
could explain the interaction between thalamic cells and inhibitory FS
cells during sensory adaptation observed by Gabernet et al., 2005
References
Bruno R. (2011). "Synchrony in sensation." Current Opinion in Neurobiology. 21, 701708.
Gabernet L, Jadhav S, Feldman D, Carandini M, and Scanziani M. (2005).
"Somatosensory Integration Controlled by Dynamic Thalamocortical Feed-Forward
Inhibition." Neuron. 48, 315–327.
Wilson H and Cowan J. (1972). "Excitatory and Inhibitory Interactions in Localized
Populations of Model Neurons." Biophysical Journal. 12.
Acknowledgements
ACKNOWLEDGMENTS
Dr. Mike Reed, program director
Duke University, Department of Mathematics
Dr. Mainak Patel, mentor
NSF