Biomedical Engineering Reference
In-Depth Information
in invertebrates and vertebrates. In order to take into account the effect of
the synapse in our model, a current such as that in (8.4) should be added to
(8.1).
The details of these models can be found in a vast literature on computa-
tional neuroscience [Koch 1999]. We would like to point out at this point that
the simplicity of the description of the dynamics that one neuron can display
(its
excitable
nature) can be misleading. Once several units are coupled, a
wide variety of complex dynamics can be found. We can couple bidirection-
ally an excitatory neuron with an inhibitory one, and oscillations can take
place [Hoppensteadt and Izhikevich 1997]. We can couple a simple neural os-
cillator to a third neuron, and a variety of subharmonic dynamics can be
found [Feingold et al. 1988, Sigman and Mindlin 2000]. It is not our inten-
tion to intimidate the reader with the complexity that can emerge out of
coupling excitable systems, but just to stress that endless complex labyrinths
can be built with these simple building blocks.
8.4 Conceptual Models and Computational Models
It is experimentally well supported that the RA-projecting neurons of HVC
burst sparsely during a motif. These units will then recruit neurons in RA,
following an architecture of connections which will, in part, be responsi-
ble for the acoustic features of the song produced [Abarbanel et al. 2004a,
Fee et al. 2004].
In Fig. 8.2, we display a schematic picture of this conceptual model. A set
of excitatory and inhibitory units represents a subpopulation of HVC neurons,
some of which project to the nucleus RA. This nucleus is represented by a set
of excitatory and inhibitory units, the excitatory ones with local connections
and the inhibitory ones with long-range connections.
One can also translate this picture into a computational model, in which
the equations described in the previous section are computationally imple-
mented in order to emulate the dynamics of the variables describing the
activities in the nuclei. In Fig. 8.3, we show an example. The time evolution
of the voltages in two RA-projecting HVC neurons is displayed in Fig. 8.3a.
These two excitatory units were connected to an inhibitory one. Unless ex-
ternally forced, these excitatory units did not fire. The connections between
these two units were such that the second unit would spike if the first one
did. Then, both units were connected to one neuron in RA, and the resulting
activity is displayed in Fig. 8.3b. This architecture allows us to reproduce ex-
perimentally observed data, in which sparse spiking activity in HVC recruits
neurons in RA.
Reproducing an experimental result is an important first step in a model,
since it builds confidence in the plausibility of the proposed mechanisms.
However, models exhibit their usefulness when they allow us to explore new
regimes not yet experimentally observed. A simulation on a larger scale could
