Biomedical Engineering Reference
In-Depth Information
2.4 SPIKINg ModElS FoR ThE NEURoN
There are two basic types of models for the fluctuations in excitability of the neuron. The so-called
conductance models that describe the channel dynamics of the ionic exchanges across the neuron
membrane and the threshold-fire models
7
that try to capture the most representative behavior of
neurons. One of the first conductance models was developed by Hodgkin and Huxley [
29
], who
described the dynamics of sodium, potassium, and leak channels in the cell membrane of the gi-
ant squid axon. Building upon the contributions of Hodgkin and Huxley, there are currently many
other more sophisticated compartmental neural models and many excellent software programs to
simulate neural excitability [
30
,
31
] and the dynamics of synapses [
32
]. One of the difficulties of this
modeling approach is that it is very computational demanding because they are physics-based mod-
eling approaches.
8
Therefore, they are mostly used to advance the study of the neuron but hardly
ever used to study neural assembly activity and its relation to behavior. A notable exception is the
Blue Brain Project being conducted at the Brain Science Institute at École Polytechnique Fédérale
de Lausanne in Switzerland by Markram et al. who teamed with IBM to build a detailed simulation
of a cortical column [
33
,
34
].
Although the Hodgkin-Huxley model provides a detailed mathematic model on the origins
of the action potential, we would like to note here that neural spikes are very stereotypical. There-
fore, in the context of motor BMIs, the shape is not the information-carrying variable. Moreover,
because of the computational infeasibility of modeling large networks of Hodgkin-Huxley neurons,
a better modeling path may be to implicitly include the ion channel dynamics and just focus on
spikes. Because organism interactions with the world are constantly changing in real time, the ques-
tion now becomes one of how can discontinuous signals (spikes) be used to represent continuous
motor control commands.
The threshold-fire model shown in Figure
2.4
captures the basic behavior of the neuron
excitability that we are describing here (i.e., the slow integration of potential at the dendritic tree
followed by a spike once the voltage crosses a threshold). In equation (
2.1
)
u
(
t
) is the membrane
potential, and
I
(
t
) is the input current. In this formulation, tau serves as a membrane constant
determined by the average conductances of the sodium and leakage channels. The appeal of this
modeling approach is that it is computationally simple, so many neurons can be simulated in a
desktop environment to simulate neural assembly behavior. The threshold-fire neuron also naturally
incorporates a number of physiological parameters, including membrane capacitance, membrane
7
The threshold-fire models are also known as the integrate and fire models.
8
The physics being referred here relates to the set of differential equations that Hodgkin and Huxley derived to
model each of the channel dynamics. It is the integration of these equations that adds to the computational complex-
ity. The simulation of even 100 Hodgkin-Huxley neurons for long periods can be a formidable task.
