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Introduction

<heat_eq_neural_sim.py> produces a spiking neural network configuration that solves a simple steady state heat equation utilizing. This uses a random walk method on a wire of length L, where walkers are allowed to move left, right, or stay in their same location in a single time step.

The resulting network can be used to benchmark neuromorphic hardware and was detailed in Smith, et al., "Solving a steady-state PDE using spiking networks and neuromorphic hardware", ICONS 2020.

The problem of this wire of length L is as follows: we attach the left end to a cooler external heat was and enforce that no heat can transfer across this left boundary (u(0) = 0, u'(0) = 0). There is a heat source nearby the wire with a given heat flux density profile.

Usage

As a script

The code can be used as a standalone script as follows:

usage: heat_eq_neural_sim.py [-h] [-L LENGTH] [-N NODES] [-w WALKERS] [-d DT] [-i [INITIAL_POSITIONS ...]] [-g GRAPH_FILE] [-m MATRIX_FILE] [-p PICKLE_FILE]

options:
  -h, --help            show this help message and exit
  -L LENGTH, --Length LENGTH
                        Length of wire
  -N NODES, --Nodes NODES
                        Number of nodes
  -w WALKERS, --walkers WALKERS
                        Number of walkers to start at each initial location
  -d DT, --dt DT        dt used by simulation
  -i [INITIAL_POSITIONS ...], --initial_positions [INITIAL_POSITIONS ...]
                        List of initial walker positions
  -g GRAPH_FILE, --graph_file GRAPH_FILE
                        Filename if you want to save the graph as yaml
  -m MATRIX_FILE, --matrix_file MATRIX_FILE
                        Filename if you want to save the matrix
  -p PICKLE_FILE, --pickle_file PICKLE_FILE
                        Filename if you want to save the graph, injection and MarkovNet as pickle

The default values for L, N, dt are those used in the paper: wire length L = 2, mesh N = 40, time step dt = 0.0001.

The matrix (Numpy ndarray) saved using MATRIX_FILE is the transition matrix if you want to implement your own network.

The graph (NetworkX DiGraph) saved using GRAPH_FILE is a yaml version of the DiGraph. See this NetworkX documentation.

As a module

Alternatively, you can get the transition matrix and graph objects directly using generate_prob_mtx and build_graph directly. Please see code and code comments for parameters and set up.

Dependencies

The code is not fully tested on other versions, but should generally be compatible.

• python >= 3.6
• numpy == 1.14.5
• networkx == 2.4
• scipy == 1.4.1

Graph and Network Details

NetworkX Graph Requirements

The network defined is a NetworkX DiGraph object. There are three main components.

1. The graph requires a property 'has_delay' with values either True (if variable delays are included) or False (if all the delays are 1).
2. Each node represents a neuron. Neuron properties should be set using node properties: 'threshold', 'potential', 'decay', 'p', 'record' (See neuron model below for full details).
3. Each edge represents a synapse. Synapse properties should be set using edge properties: 'weight', 'delay' (See synapses below).

Neuron Model

The neuron model used is generalization of a stochastic leaky-integrate-and-fire neuron. We include the following reference neuron definition, but many other compatible neuron models exist. Additionally, some other neuron models (e.g. that of TrueNorth and Loihi) can easily be adapted by simple updates to the graph.

The state is computed as follows for neuron i, where I represents an input injection, Weights are the synaptic weights and Spikes are the incoming spikes from other neurons:

1. x[i] = x[i] + I[i] + Weights[i] * Spikes
2. If x[i] > T[i], then Spikes[x_i] = (a < p[i]).
3. If x[i] <= T[i], then x[i] = x[i] * (1-m[i])

Variable Key	Definition	Variable Name	Type
potential	Internal potential value of the neuron	x[i]	float
threshold	Neuron threshold	T[i]	float
decay	Decay constant	m[i]	float
p	Probability of fire	p[i]	float [0,1]
record	List of values to record	N/A	list of strings
N/A	Random Draw	a	float [0,1]

Recording from Neurons

As referenced above, the 'record' neuron property defines which quantities should be recorded during simulation at each timestep. In this context, this value is just a note to the users and can be ignored if desired.

Synapse Model

Synapses are point synapses, where the synaptic weight is integrated into the post-synaptic neuron in a single timestep.

Delays are assumed to be integers of at least 1.

Record Key	Definition	Type
'weight'	Synaptic Weight	float
'delay'	Synaptic Delay	int [1, inf)

Injection

We represent external input into the system by an 'injection' dictionary (keys: timestep, value: array of length number of neuron). The idea is that these are integrated into the neurons directly at any timestep where it is defined. For the random walk networks, we only use input at the initial timestep, so these can be replicated with an initial potential value or incoming spikes with appropriate weighting. When exporting to yaml, the injection is stored as a graph property, i.e. graph.graph['injection']. Alternatively, it is returned as the second entry from build_graph.