Biomedical Engineering Reference
In-Depth Information
We can build a physical system displaying these timings if it is made of
two parts: one behaving as a clock, and another one that is forced by the
first one. The kind of responses that the driven system might exhibit do not
depend on its details, but on its linear/nonlinear nature.
A linear oscillating system responds to a periodic forcing in a remarkably
boring way: after a transient, the forced system ends up following the driver.
The amplitude of the response will in fact depend on how similar the natural
frequency of the system and the forcing frequency are. The phase difference
between the oscillations will depend on the parameters, but the driven system
will always end up oscillating periodically, with the same period as the driver.
Nonlinear systems, on the other hand, react in a very different way to a
periodic forcing. If the forcing frequency is similar to the natural frequency of
the system, the forced system will in fact lock itself to the driver in a period
one state. But if the difference between these frequencies is large, then the
nonlinear system will show a more complex time evolution. The nonlinear
system might display periodic behavior but the period of the solution need
not, in principle, be equal to the period of the forcing. A typical response of
a nonlinear system is to show, for wide regions of the parameter space (in
this case, the amplitude and frequency of the forcing), periodic solutions with
periods that are multiples of the driving period. This means that the system
will repeat its behavior after a time that is an integer multiple of the forcing
period.
A neuron, with no musical talent at all, will react to periodic forcing with
these kinds of solutions. The typical time it takes a neuron to return to its
rest value after an action potential defines a characteristic time. If a neuron
is periodically forced by a sequence of pulses with a period comparable to
that characteristic time, it will spike with one action potential per period of
the forcing. However, if the neuron is forced twice as fast its characteristic
time, the neuron will spike only
once
per two periods of the forcing. Two
interesting observations: it is not necessary to force the neuron exactly twice
as fast as its characteristic time to lock the neuron into a period of two. There
is a range of parameter values for which the system locks into a “period two”
solution. For the same value of the forcing amplitude, we can change the
forcing frequency within a certain range and stay locked in the same regime.
The second observation is maybe even more curious. There is a wide range
of forcing frequencies in which the neuron locks into a “period one” solution
(with
r
=
p
1
/q
1
=1
/
1). There is also a wide range of forcing frequencies
in which the neuron displays a “period two” solution (with
r
=
p
2
/q
2
=
1
/
2). Now, for a large class of nonlinear systems (excitable ones such as a
neuron among them), the largest region in frequency space between these
two frequency ranges for which a periodic solution exists is one in which a
“period three” solution occurs, characterized by
r
=
p
1
+
p
2
q
1
+
q
2
=
2
3
.
(9.1)
