# 8.6 Mathematical models of hippocampal spatial memory  (Page 6/13)

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## Calcium dependent plasticity: a better model?

We introduce Calcium Dependent Plasticity (CaDP) as a more biologically realistic model for synaptic plasticity in comparison to STDP. The model was developed by Dr. Harel Shouval of the University of Texas Medical School and colleagues at Brown University (see [link] , [link] ). All of the equations and parameter values introduced in this chapter have been taken or adapted from his 2008 publication [link] . While weight changes in the STDP model were based entirely on the order of firing and the duration of the spike interval, the synaptic weights in the CaDP model incorporate spike-timing as well as a form of rate-dependent plasticity. While STDP sets arbitrary upper bounds to achieve place field stability, CaDP can be accompanied with metaplasticity to obtain a stabilized backward shift [link] . Additionally, CaDP explicitly accounts for the interactions of biological parameters known to be involved with plasticity, including calcium, magnesium, and glutamate receptors.

## Plasticity equations

The CaDP rule is based upon a scheme where calcium levels directly affect the synaptic weights. At low calcium levels, there is no change in weight. At moderate calcium levels, we observe a depression of weights. At high calcium levels, we observe the potentiation of weights. To achieve this effect, we define a function, $\Omega$ , to have the above properties:

$\Omega \left(\left[C{a}^{2+}\right]\right)=\sigma \left(\left[C{a}^{2+}\right],{\alpha }_{2},{\beta }_{2}\right)-0.5×\sigma \left(\left[C{a}^{2+}\right],{\alpha }_{1},{\beta }_{1}\right)$

where $\sigma$ represents a sigmoid function, which we define as:

$\sigma \left(x,a,b\right)=\frac{{e}^{b\left(x-a\right)}}{1+{e}^{b\left(x-a\right)}}$

We set parameters ${\alpha }_{1}$ , ${\alpha }_{2}$ , to control the lower and upper calcium bounds of the LTD region, respectively. ${\beta }_{1}$ and ${\beta }_{2}$ adjust the concavity of the sigmoid function, where $\sigma$ approaches the Heaviside function as $\beta \to \infty$ . In our analysis of CaDP, we use the parameters $\left({\alpha }_{1},{\alpha }_{2},{\beta }_{1},{\beta }_{2}\right)=\left(0.3,0.5,40,40\right)$ .

While we might be inclined to base our entire weight change regime on this omega function, we must also realize that fluctuations in synaptic calcium levels would cause any increase in weight caused by high calcium levels to be nullified when calcium decreases into the LTD calcium concentrations. As such, we must also implement a rate function, $\eta$ , such that potentiation due to high calcium levels would outweigh the depression caused when calcium decreases through moderate concentrations and returns to equilibrium levels. Our equation for $\eta$ takes the form:

$\eta \left(\left[C{a}^{2+}\right]\right)={p}_{1}\frac{{\left(\left[C{a}^{2+}\right]+{p}_{4}\right)}^{{p}_{3}}}{{\left(\left[C{a}^{2+}\right]+{p}_{4}\right)}^{{p}_{3}}+{\left({p}_{2}\right)}^{{p}_{3}}}$

We set the parameters $\left({p}_{1},{p}_{2},{p}_{3},{p}_{4}\right)$ to $\left(2,0.5,3,0.00001\right)$ in our simulations of CaDP. As the behavior of this equation is not intuitive, we depict $\eta$ along with $\Omega$ , both as functions of calcium concentrations in figures  [link] A and  [link] B. Plasticity Equations. (A): η as a function of Calcium concentration. Notice that η monotonically increases with calcium concentrations. (B): Ω as a function of Calcium. Note that LTD occurs when 0.2< [ C a 2 + ] <0.5 and LTP occurs when [ C a 2 + ] >0.5. (C): Weight change as a function of Calcium level.

With our two calcium dependent functions, $\Omega$ and $\eta$ , we define our synaptic weight change function, depicted in [link] C as follows:

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