8.7 Mathematical modeling of hippocampal spatial memory with place  (Page 3/6)

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The cell can be modeled as a leaky capacitor that separates charge by controlling the flow of ions across the cell membrane, making a difference between potentials ${\phi }_{in}$ and ${\phi }_{out}$ across the membrane ( $v={\phi }_{in}-{\phi }_{out}$ ). This model approximates the subthreshold voltage ( $v$ before it reaches the threshold voltage ${v}_{th}$ ) and the time that $v$ does reach threshold ${v}_{th}$ . When $v$ reaches the threshold voltage, $v$ experiences a sharp increase then decrease and the cell is said to have “spiked" or “fired".

Let ${C}_{m}$ denote the membrane capacitance, ${g}_{Cl}$ and ${g}_{syn}$ denote the conductances of the chloride channels and the synapse, respectively, and ${v}_{Cl}$ denote the reversal potential (the voltage at which no net flow of chloride ions occurs). ${v}_{syn}$ is determined by the equilibrium concentrations of the ion of the associated channel, [link] . Input from other cells adds excitatory synaptic current. This input allows for the cell to depolarize and eventually reach ${v}_{th}$ and fire, which sends an electric signal to neighboring cells. The electric signal from a neighboring cell allows for voltage-gated channels to open, which affects the synaptic conductance ${g}_{syn}$ . We set the reversal potential above a threshold voltage so that as ${g}_{syn}$ increases and the channels open, the cell's voltage $v$ approaches the threshold ${v}_{th}$ . When $v\ge {v}_{th}$ , the cell fires [link] . See [link] for a model of a cell as a circuit [link] . The synaptic conductance is governed by the ODE

$\tau {g}_{syn}^{\text{'}}=-{g}_{syn}+\sum _{i}{w}_{i}^{inp}\sum _{n}\delta \left(t-{T}_{n}\right),$

where $\tau$ is the decay constant, ${w}_{i}^{inp}$ is the weight of synaptic input from the $i$ th synapse, ${T}_{n}$ is the set of input spike times for the presynaptic cell $i$ , and $\delta$ is the Dirac delta function. From Dr. Cox's book [link] , we see that applying Kirchoff's current law results in

${C}_{m}\frac{dv}{dt}+{g}_{Cl}\left(v-{v}_{Cl}\right)+{g}_{syn}\left(v-{v}_{syn}\right)=0.$

Model of dre: 120-cell ring

We use the IAF model for single cells, and we connect these cells into a ring of 120 place cells to simulate the DRE [link] . The 120-cell ring is depicted in [link] . Each cell receives external spatial input as well as input from neighboring cells. The conductances and voltages of Cells 1 and 2 are depicted in [link] as calculated by equations [link] and [link] . We monitor the weights of the connections between neighboring cells over time, where there is an arbitrary maximum weight bound so that the weights do not approach infinity and a minimum weight bound of 0 so the weights do not become negative. We also monitor how the changes of the weights affect the position of the place fields. See [link] for a depiction of the IAF model.

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