8.6 Mathematical models of hippocampal spatial memory  (Page 7/13)

where $w_i$ denotes the synaptic weight at synapse $i$ . Note that ${\left[C{a}^{2+}\right]}_i$ indicates the postsynaptic Calcium concentration at synapse $i$ . $k$ denotes a scaling factor introduced in order to generate the appropriate weight changes. In our comparison of CaDP with STDP in the next chapter, we use a value of $k$ = $\frac{1}{200}$ . It should be noted that while STDP weight updates occur following postsynaptic spikes, the CaDP regime constantly updates synaptic weights over time.

Calcium equations

As the weight changes caused by this model are dependent on Calcium, we also implement a number of equations to regulate the postsynaptic Calcium levels. Calcium level is regulated by the differential equation below:

Where ${\tau }_{Ca}$ represents the decay time constant of calcium, which we set to 50 ms. $I$ represents the Calcium influx of the synapse, which we model with the following equation:

Note that this Calcium influx value should not be associated with the synaptic input current, $I_{syn}$ . $g_N$ represents the conductance of the available N-Methyl D-Aspartate receptors (NMDARs) at the synapse (not to be confused with the excitatory conductance from our Integrate and Fire model). This can be either kept at a constant value or regulated by Metaplasticity (see "Metaplasticity" ). When not using metaplasticity, we use a value of $g_N=-1\times {10}^{-3}$ . $f(t,t_{pre})$ is the spike-timing dependent portion of input current, which we define as:

This spike-time regime consists of two components: a fast component, with a time constant of ${\tau }_f^f$ = 50 ms; as well as a slow component, with a time constant of ${\tau }_s^f$ = 200 ms. $\Theta$ represents the Heaviside function. Constants $I_f^f$ and $I_s^f$ are regulated such that $I_f^f+I_s^f=1$ . In our model, we use $I_f^f$ = .7 and $I_s^f$ = .3. Note that in this equation, the spiking is only explicitly dependent on the presynaptic firing time and not the postsynaptic firing time. Furthermore, note that unlike the STDP rule, the input presynaptic spikes are not added linearly: this spike-time equation only accounts for the most recent presynaptic spike.

$H\left(V\right)$ represents the voltage dependence of the NMDARs, which we model as:

$V$ represents the postsynaptic voltage. Note that this $V$ differs from our membrane voltage, $V_m$ , in the respect that $V$ is specific to each pre/postsynaptic pair, whereas $V_m$ is only specific to an individual place cell. We define the postsynaptic voltage as $V=V_{rest}+BPAP$ , where $V_{rest}$ is the resting potential and BPAP is the Back-propagating Action Potential, which is described later in the text. $V_{Ca}$ is the reversal potential for Calcium, which we set at 130 mV. The term $V-V_{Ca}$ is a simplified approximation of the driving force for Calcium influx. The term in the denominator is the voltage-dependance of Magnesium block on the NMDA receptor [link] . A depolarized voltage will result in the displacement of Magnesium ions from blocking NMDA receptors, allowing for more Calcium influx. We plot $f(t,t_{pre})$ and $H\left(V\right)$ in [link] .