Biomedical Engineering Reference
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where the variables ρ i stand for the input from the high vocal center, and
the τ i are time-scaling parameters.
It is important to stress that the level of the description chosen in devising
a model constrains the kind of questions that it can answer. Of course, if a
syllable in birdsong is represented by an oscillation in the average activity
of HVC, we shall not be able to study those phenomena related to how the
sparsely spiking RA-projecting HVC neurons recruit RA neurons. However, it
can warn us about global nontrivial phenomena. For example, under periodic
driving from HVC (represented by ρ i ( t )= ρ i ( t + T )), the system of (8.33)
displays subharmonic behavior. This means that under a driving from HVC
of period T , the variables E i and I might repeat themselves after a time
nT , where the value of n depends on the frequency and amplitude of the
forcing. This warns us that even at the level of telencephalic activity, diverse
patterns of activity can be generated by a unique neural substrate operated
under different conditions.
Good models should provide us with new questions (and not only good
fits to already observed data). With all its limitations, the fact that a rate
model shows this nontrivial behavior should motivate the exploration of more
sophisticated models (and, ultimately, the exploration of real systems through
experimental observations).
So far, we have discussed several levels of description of the dynamics
displayed by neural circuits. Maybe more importantly, the motivations be-
hind the studies quoted were different. Some of the efforts concentrated on
predicting the dynamics that would emerge from an anatomical substrate.
Others concentrated on the plausibility of a given mechanism.
Another interesting motivation for modeling is to understand the func-
tion of an observed feature. As an example, Fiete et al. [Fiete et al. 2004]
explored the role of the sparseness present in premotor neural codes. As we
have already discussed, the RA-projecting neurons in HVC display a sparse
bursting activity. In zebra finches, which sing motifs that consist of sequences
of different syllables, each RA-projecting neuron in HVC spikes briefly (for
about 6 ms), once per motif. Fiete and coworkers explored numerically and
analytically a network of sparsely spiking units (emulating the dynamics of
each unit by a rate model), connected to a set of units emulating the RA
nucleus. The motor output of the network was assumed to be some function
of the RA activity. With this setup, they explored the e ciency of a learning
scheme as a function of the number of bursts in HVC per motif. The learning
scheme assumed the existence of a desired motor output, and that at some
level, a network error C could be computed. With a backpropagation gradi-
ent descent rule, the changes in the synaptic weights between HVC and RA
were computed:
∂C
W ij =
∂W ij .
(8.34)
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