8.7 Mathematical modeling of hippocampal spatial memory with place  (Page 4/6)

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

where $\Delta t=t_{pre}-t_{post}$ and $A_{+}$ and $A_{-}$ scale the maximal amount of change allowed when $\Delta t$ is close to 0 [link] . $F\left(\Delta t\right)$ is depicted in [link] .

In our 120-cell model, we set a lower bound for the weights at $0mV$ and an upper bound at $5mV$ . 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 .

Model for analysis

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

where $\tau =20$ $ms$ is the membrane time constant, $v_r=-70$ $mV$ is the resting potential, $T$ is the set of input spike times, and $\delta (t-T_i)$ is the Dirac delta function. This is a simplification of equation [link] .