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MIT 9 29 - Final Project

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Gevorg Grigoryan 9.29, MIT Spring 2003 Final Project I. Introduction The goals of this project were to implement the spike-timing-dependent synaptic plasticity (STDP) model as described by L.F. Abbott et al1, quantitatively reproduce the key results presented in the paper, and to use the model for predicting the stimulus timing-dependent plasticity in orientation selective cortical cells experimentally observed by Yang Dan et al2. II. Methods The spike-timing-dependent synaptic plasticity model was implemented exactly as outlined in the methods section of L.F. Abbott et al1. Briefly, 1000 excitatory and 200 inhibitory neurons all firing at the same average rate were modeled as inputs into a single integrate-and-fire neuron governed by the equation ( )( ) ( )( )VEtgVEtgVVdtdVininexexrestm−+−+−=τ, where ()()tgtginex and , were the current excitatory and inhibitory conductances, and inexEE , were the Nernst potentials of excitatory and inhibitory synapses. The peak conductances of excitatory synapses were modified according to the rules described in the paper (basically, synapses receiving an action potential just before the post-synaptic action potential were rewarded and those receiving an action potential just after the post-synaptic action potential were decreased in weight). Peak excitatory conductances were not allowed to drop below zero or incrase above maxg . The peak conductances of inhibitory synapses were constant (and the same for all inhibitory synapses). Whenever a synapse received an action potential,()tgex or ()tgin was be incremented by the peak conductance of the synapse. ()tgex and ()tgin also exponentially decayed with time. Only the case with uncorrelated Poisson spike train inputs was considered. Dan et al2 proposed a model for their observed stimulus timing-dependent plasticity. The model is outlined in Figure 1. Figure 1: taken from Figure 1 of Dan et al2.The idea is that if there are connections between the monitored orientation selective neuron (neuron a with optimal orientation taken to be 0°) and other orientation specific neurons (for example neuron b with optimal orientation +α), then continually presenting the subject with a grating at an angle α quickly followed by a grating at an angle 0° will cause the synapse between neurons a and b to strengthen and hence will cause a shift in the orientation tuning curve of neuron a towards that of neuron b. I used this scheme as the basis for my model, which is represented in Figure 2. Figure 2: my model for stimulus timing dependent plasticity. The central neuron is modeled explicitly as an integrate-and-fire neuron. The orientation specificity is introduced through an orientation specific conductance org (which is perhaps the total conductance coming from an array of center surround neurons). So the equation governing this neuron is ( )( ) ( )( )VEgVEtgVVdtdVororexexrestm−+−+−=ατ, where α is the angle of the grating being currently presented. ()tgex is the total synaptic conductance coming from the connections to other orientation selective neurons. These are modeled implicitly as two neurons c and b, one with optimal orientation at Lα (I used -15°) and the other at Rα (+15°). No inhibitory synapses were included in the model. In their paper Dan et al showed that the orientation tuning curves of the neurons they studied were gaussian. So in order to model that kind of behavior, I had to solve for the form of ()αorg that would produce a gaussian orientation tuning curve with the peak firing rate and standard deviation like those observed in experiments presented in the paper. I picked Figure 3A from the paper as my target behavior (see Figure 3 below), so I wanted my neuron a to have a peak firing rate of about 4 spikes per second and the standard deviation of the orientation tuning curve to be abound 15°. a cs cs cs cs cs cs gor }c b 0° αL αRFigure 3: taken from Figure 3 of Dan et al2. Assuming the above voltage equation, the firing rate of such a neuron as a function of grating angle α is: ( )()() −−++=∞∞VVVVggromorexθτααln1, where θV is the threshold voltage for firing, oV is the reset voltage after an action potential has been fired, and ()oexororexexrestggEgEgVV++++=∞1α. In order to have this firing rate to be a gaussian function, we have to solve ()()( )222maxln1σααθτα−−∞∞= −−++erVVVVggomorex with or 15 and Hz, 4max==σ. Because it is impossible to solve this analytically, I solved it numerically using the steepest descent method (since the derivative of the firing rate with respect to α can be calculated analytically). This created a lookup table for org values depending on α. The solution (for the parameters I used) is shown in Figure 4 and the resulting firing rate of the neuron is shown in Figure 5 (note, I assumed that prior to conditioning there were no connections between neuron a and neurons b and c, so for solving the above equation I used gex = 0). The firing rate was calculated as an average of 200 spikes.Figure 4: gor as a function of orientation. Figure 5: firing rate of neuron a as a function of orientation (before conditioning). Once gor was solved for, the system was put through conditioning, which reflected the conditioning experiment carried out in the paper by Dan et al. The scheme of their experiment is presented in Figure 6. My implementation essentially followed part A of the figure exactly. First, α was set to -15° (the optimal orientation for neuron c). After 8.3 ms, the neuron was put into the rest state by “removing” the grating (i.e. the ()()VEgoror−α term was removed from the voltage equation). After another 8.3 ms, αwas set to 0° (the optimal orientation for neuron a). And finally, after 8.3 ms the neuron was put into the rest state for 100 ms. This was repeated many times. The authors used several different interval lengths between the showing of the two gratings (8.3 ms to 41.5 ms), but I only tried one interval of 8.3 ms. A similar conditioning was performed where first neuron b was excited (i.e. and α is set to +15° ) then neuron a. Figure 6: taken from Figure 2 of Dan et al2. After the conditioning for 200 seconds of simulated time (they used 3 minutes in the paper), the orientation tuning curve of neuron a was calculated in the same manner as before (averaged from 200 spikes). For the period of orientation tuning


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MIT 9 29 - Final Project

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