8.6 Mathematical models of hippocampal spatial memory  (Page 5/13)

$A_{+}$ and $A_{-}$ represent coefficients for the maximum values of potentiation and depression per spike pair, respectively. We set $A_{+}$ = .4 and $A_{-}$ = .42. ${\tau }_{+}$ and ${\tau }_{-}$ are the time constants for LTP and LTD, both of which are set to 20 ms. We depict a plot of weight changes versus spike times below.

The rule is fairly intuitive: if the presynaptic cell fires before the postsynaptic cell, the weight increases. If the postsynaptic cell fires before the presynaptic cell, the weight decreases. The closer the spike times are, the greater the magnitude of the weight change. As this weight change model is only dependent on spike times, more spike pairs will result in more weight changes. As such, the weights would be able to increase without bound. To prevent excessive weight increases, we impose a maximum weight, which we denote $W_{max}$ , of 5 in our simulations involving STDP. We also set a lower weight limit at zero, ensuring no negative synaptic weights. A more detailed analysis of STDP dynamics can be found in Georgene Jalbuena's report of Spike-time Dependent Plasticity.

There exist a number of variations on this plasticity rule. Some adjust synaptic weights based on the spike times of the most recent spike pair, whereas others account for all of the pre-postsynaptic spike pairs. We will not discuss this in-depth: we find that the weight changes associated with accounting for all spike pairs are slightly increased, but with very little difference besides this. Since we scale the degree of weight change with $A_{+}$ and $A_{-}$ , we will base our analysis of STDP off of results with weight changes involving only the most recent spike pair.

A variant of STDP that we do consider, known as Multiplicative STDP, proposes a more feasible means of synaptic weight changes by incorporating the current weight into the weight change equation (see [link] ). We describe this weight modification scheme as $dW=g\left(\Delta t\right)$ , where:

This version of STDP allows for a more asymptotic approach to the weight bounds. Additionally, it produces a behavior more similar to the experimental results, which showed that more marked changes in synaptic weights occur in the first few laps and become lessened in later laps.

The STDP model is currently implemented in the Double Rotation experiment model. It reproduces the backward shift and results in final place field stability, as found in the results of the experiment. While it has been successful in reproducing experimental results, the model itself is still flawed, as it requires an arbitrary upper limit to be set on the maximum synaptic weight in order to achieve place field stability. As such, STDP may not appear to be the most biologically realistic model for modeling synaptic plasticity. We explore a newer alternative to STDP, coined as Calcium Dependent Plasticity, in the next chapter.