Integration of Rule-Based Models and Compartmental Models of Neurons

Abstract

Synaptic plasticity depends on the interaction between electrical activity in neurons and the synaptic proteome, the collection of over 1000 proteins in the post-synaptic density (PSD) of synapses. To construct models of synaptic plasticity with realistic numbers of proteins, we aim to combine rule-based models of molecular interactions in the synaptic proteome with compartmental models of the electrical activity of neurons. Rule-based models allow interactions between the combinatorially large number of protein complexes in the postsynaptic proteome to be expressed straightforwardly. Simulations of rule-based models are stochastic and thus can deal with the small copy numbers of proteins and complexes in the PSD. Compartmental models of neurons are expressed as systems of coupled ordinary differential equations and solved deterministically. We present an algorithm which incorporates stochastic rule-based models into deterministic compartmental models and demonstrate an implementation (“KappaNEURON”) of this hybrid system using the SpatialKappa and NEURON simulators.

The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement nos. 241498 (EUROSPIN project), 242167 (SynSys-project) and 604102 (Human Brain Project). We thank Anatoly Sorokin for his help with SpatialKappa and comments on an earlier version of the manuscript, and Vincent Danos for thought-provoking discussions.

A Appendix

1.1 A.1 Kappa Simulation Method

To understand the asynchronous nature of the Kappa simulation method, we first illustrate Gillespie’s direct method [10] by applying it to the kinetic scheme description of a calcium pump shown in Eq. (7). Here Ca represents intracellular calcium, P represents a pump molecule in the membrane, is the pump molecule bound by calcium and $k_1$ and $k_2$ are rate coefficients, which are rescaled to the variables $\gamma _1$ and $\gamma _2$ as explained in Sect. 2.5. To apply the Gillespie method to this scheme:

• 1. Compute the propensities of the reactions and

• 2. The total propensity is $A = a_1+a_2$

• 3. Pick reaction $\mathrm {R}_i$ with probability $a_i/A$

• 4. Pick time to reaction $T = -(\ln r)/A$, where r is a random number drawn uniformly from the interval (0, 1).

• 5. Goto 1

Kappa uses an analogous method, but applied to rules that are currently active. Both methods are event-based rather than time-step based.

1.2 A.2 Justification for Throwing Away Events

To justify throwing away events occurring after a time step ending at $t+\varDelta t$, we need to show that the distribution of event times (measured from t) is the same in two cases:

• 1. The event time T is drawn from an exponential distribution $A\exp (-AT)$ (for $T>0$), where A is the propensity.

• 2. An event time $T_0$ is drawn as above. If $T_0<\varDelta t$, accept $T=T_0$ as the event time. If $T_0\ge \varDelta t$, throw away this event time and sample a new interval $T_1$ from an exponential distribution with a time constant of A, i.e. $A\exp (-AT_1)$. Set the event time to $T=\varDelta t + T_1$.

In the second case, the overall distribution is:

$$\begin{aligned} \begin{aligned} P(\text{ event } \text{ at } T < \varDelta t)&= A\exp (-AT) \\ P(\text{ event } \text{ at } T \ge \varDelta t)&= P(\text{ survival } \text{ to } \varDelta t) P(\text{ event } \text{ at } T_1) \\&= \exp (-A\varDelta t) A\exp (-A(T-\varDelta t)) \\&= A\exp (-AT) \end{aligned} \end{aligned}$$

(18)

Here we have used $T_1 = T - \varDelta t$. Thus the distributions are the same in both cases.