Equation-oriented specification of neural models for simulations

Abstract

Simulating biological neuronal networks is a core method of research in computational neuroscience. A full specification of such a network model includes a description of the dynamics and state changes of neurons and synapses, as well as the synaptic connectivity patterns and the initial values of all parameters. A standard approach in neuronal modeling software is to build network models based on a library of pre-defined components and mechanisms; if a model component does not yet exist, it has to be defined in a special-purpose or general low-level language and potentially be compiled and linked with the simulator. Here we propose an alternative approach that allows flexible definition of models by writing textual descriptions based on mathematical notation. We demonstrate that this approach allows the definition of a wide range of models with minimal syntax. Furthermore, such explicit model descriptions allow the generation of executable code for various target languages and devices, since the description is not tied to an implementation. Finally, this approach also has advantages for readability and reproducibility, because the model description is fully explicit, and because it can be automatically parsed and transformed into formatted descriptions. The presented approach has been implemented in the Brian2 simulator.

Keywords: computational neuroscience; neuroscience; python; simulation; software.

Figure 1

Examples for neuronal model descriptions in Brian2. (A) Differential equations and parameters are defined via multi-line strings, units are specified after the colon. Note that the units specify the units of the variables defined in the respective line which in the case of differential equations is not equivalent to the units of left-hand side and right-hand side of the equations (volt vs. volt/second). (B) The symbol xi is used to refer to a stochastic variable for defining stochastic equations. (C,D) Threshold conditions and reset statements are also defined by strings.

Figure 2

Examples for refractoriness formulations in Brian2. (A,B) Refractoriness condition based on time since last spike. (C) Refractoriness condition based on membrane potential threshold crossing. (D) Modifying differential equations based on refractoriness state, the keyword unless refractory clamps a variable x during the refractory period, i.e., $\frac{\text{d}x}{\text{d}t}=0$.

Figure 3

Examples for synaptic model descriptions in Brian2. (A–C) Forward propagation is described by specifying statements to be executed on the arrival of a pre-synaptic spike (keyword argument pre), changing the value of post-synaptic variables (suffix _post). (D,E) Equivalent definitions of spike timing-dependent plasticity using forward and backward propagation. The (event-driven) at the end of the differential equations in (D) makes Brian2 automatically convert the equations into the alternative formulation given in (E). The name w_max refers to a constant defined outside of the synaptic model. clip is a function that restricts the values given as the first argument to the range specified by the second and third argument.

Figure 4

Examples for synaptic model descriptions involving summed variables in Brian2. (A,B) The equation lines ending with (summed) specify that the post-synaptic variable indicated with the suffix _post is set by summing the corresponding synaptic values on each time step.

Figure 5

Examples for synaptic connection descriptions in Brian2. Strings define conditions based on the pre-synaptic neuron index i, post-synaptic neuron index j, and pre- and post-synaptic neuron variables with suffix _pre and _post. Optionally, a value or expression for the probability of creating a synapse can be given as the keyword argument p. The number of synapses to create for each connection can be given as the keyword argument n.

Figure 6

Examples for specifying initial state variable values in Brian2. (A) Neuronal variables. (B–D) Synaptic variables based on pre- and post-synaptic neuronal variables indicated with suffix _pre and _post respectively.

Figure 7

The code-generation process. Left column: user level descriptions are given either as a set of equations with a numerical integration method, or directly as a sequence of mathematical statements. Middle column: in both cases the user level description is turned into a sequence of mathematical statements which we refer to as “abstract code”. This involves no change in the case that the user provided a sequence of statements. Right column: the sequence of abstract code statements is converted into valid code in the target language.