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Phase Delayed Inhibition in the Rat Barrel Cortex Runjing (Bryan) Liu; Dr. Mainak Patel Department of Mathematics, Duke University. Abstract . Results. Tailoring the model to fit the phase delayed inhibition network, we obtain: .

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Phase Delayed Inhibition in the Rat Barrel Cortex

Runjing (Bryan) Liu; Dr. Mainak PatelDepartment of Mathematics, Duke University



Tailoring the model to fit the phase delayed inhibition network, we obtain:

  • 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.

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.

Insert plot here

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, τiand τe are constants, and P(t) is the stimulus received by the encoder neuron population. δiandδe are sigmoid functions of the form:

Figure 6: trajectories at two different synchrony values

Figure 5: A phase plane and the nullclines at a fixed time t

  • 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

δiandδe 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).



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:

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;

Figure 8: effects of changing synaptic strengths

  • 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 cannot fire.


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).

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 Gabernetet al., 2005

Figure 3 the effect of synchrony on decoder activity

Figure 2: P(t) at three different levels of synchrony

Figure 7: phase delayed inhibition in the barrel cortex

Sensory Adaptation

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 Gabernetet 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 1: The basic phase delayed inhibition network


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)

  • 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.

Bruno R. (2011). "Synchrony in sensation." Current Opinion in Neurobiology. 21, 701-708.

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.

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

Mathematical Model: Wilson-Cowan

  • 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

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).



Dr. Mike Reed, program director

Dr. Mainak Patel, mentor

Duke University, Department of Mathematics