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Depiction of the IAF model for Cell 1 and Cell 2 from [link] . We calculate the conductances and voltages for Cells 1 and 2 by [link] and [link] , respectively. v t h = - 54 m V , and w e x t = w 12 = 2 S are the weights of the synapses from the external source and Cell 1, respectively. It can be seen that in Place Field 1 ( 0 < t < 100 ), Cell 1 receives external input every 20 ms (in the form of conductance spikes), and in Place Field 2 ( 100 t < 200 ), Cell 2 receives the external input. At each external input spike, the corresponding cell fires (its voltage spikes) and gives a small amount of input to its neighboring cells. ( IAF2cellsweight.m )

Stdp

To determine how the weights of the synapses change, we use an STDP model [link] . The spikes times of the pre- and post-synaptic cells are compared, and the smaller the time difference, the more the weight is adjusted. See [link] . The percentage of weight change is determined by

F ( Δ t ) = A + e Δ t / τ + , Δ t < 0 - A - e - Δ t / τ - , Δ t 0 ,

where Δ t = t p r e - t p o s t and A + and A - scale the maximal amount of change allowed when Δ t is close to 0 [link] . F ( Δ t ) is depicted in [link] .

Percentage of weight change due to STDP. This graph shows that as Δ t = t p r e - t p o s t approaches 0, the percentage of change in the weight of the synapse increases [link] . F ( Δ t ) is calculated by Equation [link] .

In our 120-cell model, we set a lower bound for the weights at 0 m V and an upper bound at 5 m V . The necessity for an upper weight bound is one of the weaknesses of the STDP model, so Andrew Wu, another member of this PFUG, has done work with other plasticity models. See the link "Mathematical Models of Hippocampal Spatial Memory".

The work with the 120-cell model is all computational. Our results show that each lap, the weight of the synapse from Cell 1 to Cell 2 ( w 12 ) increases toward a set weight bound and the weight of the synapse from Cell 2 to Cell 1 ( w 21 ) decreases to 0 as in [link] . We also see that after 4 laps around the path, the place fields start to shift backward. [link] shows that the spike time of Cell 2 decreases each lap, which is indicative of a backward shift of Cell 2's place field. See IAF120cells1stspks.m .

Weights of synapses over time. The weight of the synapse from Cell 1 to Cell 2 ( w 12 ) increases each lap (1 lap = 12,000 ms) as the weight of the synapse from Cell 2 to Cell 1 ( w 21 ) decreases each lap to 0. ( IAF120cellsSTDP.m )
Backward shift of place field of Cell 2 in m s . The rat spends 100 m s in each place field. It can be seen here that after as few as 4 laps around the track, place cell 2 starts to fire earlier and its place field shifts backwards. ( IAF120cells1stspks.m )

Model for analysis

Simple 1-cell system. Cell 1 receives input of weight w i n p at an interspike interval I .

We begin by considering only one place cell which receives input from one external source with constant weight w i n p at a set interspike interval, I , as depicted in [link] . The following equation gives the voltage v in m V of the cell at time t with n total input spikes:

τ v ' ( t ) = ( v r - v ( t ) ) + w i n p i = 1 n δ ( t - T i )

where τ = 20 m s is the membrane time constant, v r = - 70 m V is the resting potential, T is the set of input spike times, and δ ( t - T i ) is the Dirac delta function. This is a simplification of equation [link] .

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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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