8.6 Mathematical models of hippocampal spatial memory  (Page 8/13)

To regulate postsynaptic voltage, we use a Back-propagating Action Potential (BPAP), which increases whenever the postsynaptic cell fires. The BPAP represents the postsynaptic cell's feedback, indicative of how recently the postsynaptic cell has spiked. The magnitude of the BPAP is dependent on the following equation:

Similar to $f$ , the decay of the BPAP is regulated by a fast and a slow component. Upon a postsynaptic spike the BPAP causes the voltage to reach a maximum of 100 mV above the resting potential. We allow a delay of 2 ms between postsynaptic firing and the delivery of the BPAP to the dendrites. The fast component has time constant ${\tau }_f^b$ = 3 ms and the slow component has time constant ${\tau }_s^b$ = 25 ms. $I_f^b$ and $I_s^b$ are also chosen such that they add up to one: here, we use $I_f^b$ = 0.75 and $I_s^b$ = 0.25. Note that whenever a cell fires, the BPAP is sent through all dendrites, increasing the voltage at all of its presynaptic connections. In this plasticity model, the BPAP provides postsynaptic feedback and drives calcium influx, which allows for the potentiation of synaptic weights. Along with the function $f$ , the BPAP ensures Calcium dependence on both pre and postsynaptic spike times.

Metaplasticity

To ensure the stabilization of synaptic weights after several laps around the track, we use metaplasticity to limit NMDAR conductance ( $g_N$ ) after repeated high-frequency postsynaptic stimulation. Metaplasticity follows a voltage-dependent kinetic model of NMDAR insertion and removal from the synapse as prescribed by the equation below:

$a$ is a scaling factor used to control the rate of change in NMDAR conductance. $k_{+}$ is the insertion rate of unused NMDA receptors into the synapse, which we set at $8\times {10}^{-5}$ . $k_{-}{(V-V_{rest})}^n$ is the removal rate of NMDA receptors from the synapse: we use $k_{-}$ = $8\times {10}^{-7}$ , $n$ = 2, and $V_{rest}$ as the cell resting potential. Like our voltage-dependent H-function, $V$ in the NMDAR conductance equation represents the postsynaptic voltage (as opposed to the membrane voltage). $g_t$ is the maximum value for NMDAR conductance, which we set at $-1\times {10}^{-3}$ . Note that all NMDAR conductance values are negative. We depict equilibrium NMDAR conductances as a function of voltage in [link] below.