Individual Neurons - Computational Explorations in Cognitive Neuroscience - page 44

1−

where N spikes is the number of spikes fired during that
time period, N cycles is the total number of cycles, and

Rate Code

−30−

Spike

−40−

0.5−

eq ( eq_gain in the simulator) is a gain factor that
rescales the value to better fill the 0 to 1 range used
by the rate code activations (the result is also clipped
to ensure it stays within the 0 ￿ 1 range). The neuron
might actually have something like this rate-code equiv-
alent value in the form of the concentration of internal
calcium ions, which may be what learning is based on
(more on this in chapters 4-6).

−50−

act

0−

−60−

V_m

−0.5−

−70−

−80−

−1−

0 5 10 15 20 25 30 35 40

Figure 2.12: Simple discrete spiking output, showing
that as the membrane potential ( V m ) exceeds the threshold
( ￿55mV ), the activation output ( act ) goes to 1. On the next
time step, the membrane potential is reset to a hyperpolarized
value, and the activation returns to 0. Also shown for compar-
ison is the equivalent rate-code output activation (dotted line)
that would be produced by this level of input excitation.

2.5.4 The Rate Code Output Function

As we have stated, a rate code output function can
provide smoother activation dynamics than the discrete
spiking function, and constitutes a reasonable approx-
imation to a population of neurons. To compute a
rate code output, we need a function that takes the
membrane potential at the present time and gives the
expected instantaneous firing rate associated with that
membrane potential (assuming it were to remain con-
stant). Because there is no spiking, this membrane po-
tential is not reset, so it continuously reflects the balance
of inputs to the neuron as computed by the membrane
potential update equation.
In the simple discrete spiking mechanism described
in the previous section, the main factor that determines
the spiking rate is the time it takes for the membrane
potential to return to the threshold level after being re-
set by the previous spike (figure 2.12). Although we
were unable to write a closed-form expression for this
time interval as a function of a non-resetting membrane
potential, simulations reveal that it can be summarized
reasonably accurately with a function of the X-over-X-
plus-1 (XX1) form (suggested by Marius Usher in per-
sonal communication):

K + ions). A set of detailed mathematical equations de-
rived by Hodgkin and Huxley (1952) describe the way
that the two voltage-gated channels open and close as
a function of the membrane potential. However, these
equations provide more detail than our model requires.
Instead, we use a simple threshold mechanism that
results in an activation value ( act in the simulator)
of 1 if the membrane potential exceeds the threshold
( ￿ , thr in the simulator), and a zero otherwise (fig-
ure 2.12). On the time step following the spike, the
membrane potential is reset to a sub-resting level (de-
termined by parameter v_m_r in the simulator), which
captures the refractory effect seen in real neurons. Also,
to simulate the temporally extended effects that a single
spike can have on a postsynaptic neuron (e.g., due to
extended activation of the receptors by neurotransmit-
ter and/or delay in receptor closing after being opened)
the spike activation can be extended for multiple cycles
as determined by the dur parameter.
For ease of comparison with the rate-code function
described next, and to enable learning to operate on
continuous valued numbers, we also need to compute
a time-averaged version of the firing rate, which we call
the rate-code equivalent activation ( y eq
j

(2.19)

where y j is the activation ( act in the simulator), ￿
is again the threshold value, ￿ is a gain parameter
( act_gain in the simulator), and the expression [x]+
means the positive component of x and zero if negative.
Interestingly, equation 2.19 is of the same general
form as that used to compute V m itself (and can be given

, act_eq in the
simulator). This is computed over a specified period of
updates ( cycles ), as follows:

(2.18)