Neuronal Model

John Matthias

The majority of the processing of sensory information in our brains is thought to take place in the Cerebral Cortex, which contains a population of billions of neurons, each making thousands of synaptic connections with its neighbours. A single neuron can be thought of as a cell which generates a travelling spiking signal to these connected neighbours when the voltage on its membrane exceeds a certain threshold voltage, a process which is called 'firing'. A neuron receiving several spikes simultaneously (or within a very small time-window) is likely to have its voltage pushed beyond the threshold level and will therefore in turn, send spike signals to its connected neighbours. Furthermore, connections between most neurons which cause spiking signals tend to become potentiated and those which do not become depleted, a phenomenon known as 'Synaptic Plasticity'. The dynamics of millions of such adaptive, interconnected neurons thus provides an extremely rich behaviour, especially on a collective level of description [For a comprehensive introduction, see for example Gerstner and Kistler 2002] and patterns of firing regularly occur in groups of neurons. The following diagram, which is taken from a mathematical simulation of a group of 1000 coupled neurons [Izhikevich et.al. (2004)] shows an example of such collective firing behaviour.

The neurons are numbered on the ordinate y-axis (with neuron number 1 at the bottom, and neuron number 1000 at the top) and the time, which runs from zero to 1000 milli-seconds (or one second), is on the x-axis. This is therefore a simulation of one second's activity of this group of 1000 artificial neurons. Every time a neuron fires a blue dot is placed on the graph at the appropriate time on a line horizontally drawn from that particular neuron. The dots on the graph can thus be regarded as firing 'events'. In the particular graph shown, because of the plasticity of the neural connections, many of the events are centred in four bands, which appear as a pulse or 'wave' of spiking events in real-time.

Graph The spiking events are indeterminate (not predictable in advance) but are certainly not randomly distributed, and, as is the case with the above scenario, can be very correlated. A rhythmic pattern such as the one pictured above is very likely to be connected with the 'polychronous' firing of a particular group of neurons [Izhikevich et.al. (2004)], in which the firing of a particular neuron generates a sequence of events which stimulate a large number of neurons, which form a closed group with the connections between these neurons being reinforced through repeated firing of the first neuron (the group of neurons fire not in synchrony but with polychrony). It is the musicality of these sequences of firing events, or rhythms, which we are very interested in. In our research (see for example, Miranda and Matthias (2005), Grant et. al. (2007), Matthias and Ryan (2008)) we have started to look at what happens when each firing event, or dot in a diagram like the one above is represented by a sonic event.

The Fragmented Orchestra

In The Fragmented Orchestra, each dot is represented by a tiny sample of sound streamed from one of the 24 sites. The cortex is tiny, consisting of just 24 neurons, but is sufficiently large to achieve many complex rhythms and collective firing behaviours. Each site is represented by its own neuron, which is also stimulated by the sound from that site. The samples of sound are relayed in real-time and last between 30 milliseconds and 0.5 seconds. We have adapted a mathematical model of biological neuronal networks developed by Eugene Izhikevich and others [Izhikevich et.al. (2004)], which is one of many models of spiking neuronal networks which are based on a simplification of a model developed by Hodgkin and Huxley in the 1950s [Hodgkin and Huxley (1952)]. These models are essentially electrical in nature and consider the relationships between the flow of ions across cell membranes and the flow of voltage signals between the cells. In a sense, we have created a strange, hybrid organism, involving 24 cortical neurons which are also all crude sensory neurons (they are stimulated by the 'volume' of the sound). This tiny organism is then spread across the 24 sites and online at this website and turned into a musical instrument...

John Matthias
Physicist, musician and artist, The Fragmented Orchestra


references

Izhikevich E.M., Gally J.A. and Edelman G.M. 2004 Spike-timing dynamics of neuronal groups. Cerebral Cortex 14 933-944
Gerstner W. and Kistler W (2002) Spiking Neuron Models: Single Neurons, populations and plasticity. Cambridge University Press
Miranda, E. R. and Matthias, J.R. (2005). Granular Sampling using a Pulse-Coupled Network of Spiking Neurons Proceedings of EvoWorkshops 2005, Lecture Notes in Computer Science 3449, pp. 539-544. Berlin: Springer-Verlag.
John Matthias and Nick Ryan 2008. Cortical Songs, Nonclassical Records (CD)
Hodgkin, A. L. and Huxley, A. F. (1952). A quantitative description of ion currents and its applications to conduction and excitation in nerve membranes. J. Physiol. (Lond.), 117:500-544.