Dynamical Systems and Silicon Based Hybrid Spiking Neurons

Large scale brain models may be run using limited computing resources by using hybrid spiking models rather than conductance based models (for examples see my blog posts referenced below). Hybrid spiking models combine a smooth spike-generation mechanism but they do after-spike reset of neuron membrane potential rather than use channel conductance algorithms. Rather than mimic the details of channel conductance, hybrid spiking models mimic the dynamics of individual neurons.

The paper “Hybrid spiking models” by Dr. Eugene M. Izhikevich was published November 13, 2010 in Philosophical Transactions of the Royal Society A reviews the advantages of using hybrid spiking models and describes a way to implement these models so that they’re optimized for embodiment in computer hardware.

A major problem when implementing conductance based neuron models is deciding the values of the large number of variables and parameters that may, in principle, be directly measured but are actually difficult to impossible to measure. Even if this issue were to be overcome, conductance based neuron models are computationally intensive. In contrast, hybrid spiking neuron models are based on the dynamics of the neuron and generally reduce to just four parameters.

What is meant by neuron dynamics? The typical time (x-axis) versus membrane potential (y-axis) representation of an action potential (or spike) is shown in Figure 1. In contrast, Figure 2 shows the dynamical systems representation of action potentials. Look, for instance, at Figure 1b and the line at bottom labeled “input.” Current is injected into the neuron where you see input jump up. Above this you can see the neuron’s membrane potential increase.

The equivalent phenomenon in the dynamical system representation in Figure 2a is the solid bold line inside the shaded area. The black dot denotes an area of stable equilibrium known as an attractor. As current is injected into the neuron its membrane potential increases (the bold line move to the right along the x-axis) but as long as the injected current pushes the potential to a level that is inside the gray box the membrane potential spontaneously returns back to equilibrium (the attractor). However, if the membrane potential is pushed beyond the gray box (beyond threshold) the membrane potential spontaneously follows the bold line that moves outside the gray box and circles counter clockwise around the phase plane. This traces the generation of the action potential and then the movement of the membrane potential back to equilibrium.

Note: Play with a simple model of spiking neurons using a freely available MATLAB program based on Eugene Izhikevich’s November 2003 paper “Simple Model of Spiking Neurons” published in IEEE Transactions on Neural Networks. Access the paper, MATLAB program, and other information here.

Hybrid models of spiking neurons have enabled very large scale simulations of brain function including a simulation that encompassed 22 different neuronal types, 10 to the 11 neurons (a one followed by eleven zeros), and almost one quadrillion synapses. It’ll be interesting to see their capabilities when they are implemented in computer hardware.

Note: For those interested in learning more about a dynamical systems approach to understanding neuron activity, I highly recommend Dr. Izhikevich’s 2007 book Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (Computational Neuroscience).

Other related blog posts:

Memory and the Precise Timing of Signals in the Brain

Neuronal Group Selection and Spike Timing Dependent Plasticity