176 CATALYZING INQUIRY

express quantitatively the feedback entailed in the relationship between changing ionic flows and
changing membrane potential. Originally based on data collected from experiments on the giant axon of
the squid, the physical model used is that of a membrane separating two infinite regions, each of which
is homogeneous on its side of the membrane.
In the nervous system, different kinds of ions pass through the membrane, and the flow of ions
through these channels is voltage dependent. In the model, a circuit is used to represent the ion flows
and potential differences that drive ion flow. The semipermeable cell membrane separating the interior
of the cell from the extracellular liquid is modeled as a capacitor, and each ion channel is modeled as a
separately variable resistor. In series with each variable resistor is a battery representing the Nernst
potential arising from the difference in ion concentration on each side of the membrane. All of these
components are connected in parallel and are driven by a time-varying current source to ground. If a
time-varying input current is injected into the cell, it may add further charge on the capacitor, or the
added charge may leak through the channels in the cell membrane. Because of active ion transport
through the cell membrane, the ion concentration inside the cell is different from that in the extracellular
liquid. The potential generated by the difference in ion concentration is represented by a battery.
Elementary circuit theory allows the construction of a set of differential equations relating the
different ion currents to the potential difference across the membrane. Using this set of differential
equations, certain essential features of neural behavior can be modeled. For example, assuming appro-

PSCs synaptic weights^

decoding

recurrent connections

neuron

soma

dendrites

input

spikes

h (^) syn( s )
h (^) syn( s )

recurrent

matrix

output

spikes

encoding

dynamics

matrices

FIGURE 5.13 A generic neural subsystem. The diagram depicts the mathematical analysis of a neural subsystem
and its mapping onto the biological system—a population of neurons. Labels outside the gray boxes indicate the
relevant biological structures and processes. Neural action potentials (spikes) coming from a previous neural
population generate weighted post-synaptic currents (PSCs) in the dendrites of the neurons to which they are
connected. The subsequent voltage changes travel to the neural somata, where action potentials are generated,
resulting in output spikes. Because the input and output are neural spikes, this kind of subsystem can be linked to
others like it, permitting the construction of larger, more complex neural circuits (see Figure 5.15 for an example).
Note that labels inside the gray boxes are generated based on understanding of the purpose of the neural system
being modeled and on current understanding of neural representation (encoding), computation (decoding), and
dynamics (dynamics matrices and hsyn). Building simulations using these methods leads to a better understanding
of how neural systems perform the complex functions they do. SOURCE: Courtesy of Chris Eliasmith, University
of Waterloo.