Biomedical Engineering Reference
In-Depth Information
78 CHAPTER 7. NETWORKS OF NEURONS
7.1.2 Conduction down the Axon and Synaptic Connections
The axon of the Pyramidal cell is not modeled directly because it is assumed that any action potential at
the soma will reach the axon terminal after some delay. The axon is therefore modeled simply as a delay
that depends upon the conduction velocity and length of the axon.
Once an action potential is delayed and reaches the pre-synapse, the impact on the post-synapse
is modeled using the generic synaptic current equation first introduced in Eq. (6.3).
c
e
O(t)
V
post
E
e
I
syn
(t)
=
−
(7.1)
m
where the maximum conductance term is
c
e
and the gating variable,
O
, takes the form
dO
dt
=
rect
(t
on
,t
off
)
−
O.
(7.2)
The rect function is a pulse that is equal to 1 between
t
on
and
t
off
and 0 everywhere else. In Traub's model,
if the soma voltage is greater than 20
mV
and hasn't fired in 3
msec
(i.e., refractory period), then
t
on
is
set to the delay for the axon.
t
off
determines how long the pulse will last and is governed by the type of
synapse.
7.1.3 Interneurons
Real networks of neurons are rarely composed of a single cell type. Traub's group therefore incorporated
interneurons
, also known as inhibitory neurons, into their model by making three changes to the Pyramidal
cell model. First, the multicompartment geometry was reduced. Second, the active membrane at the soma
was changed to eliminate the Calcium current. Third, synaptic connection from interneurons were made
to be inhibitory by changing the Nernst potential and
t
off
in Eqs. (7.1) and (7.2) at the post-synapse. The
strength of the inhibitory connection was called
c
i
.
7.2 MODEL BEHAVIOR
Traub's research lab was able to simulate nearly 10,000 individual cells and made some ground-breaking
observations that agreed well with experimental recordings. Before explaining the large-scale trends, they
demonstrated three interesting phenomenon, in networks of two or three cells, that formed the basis for
the more complex behavior. We repeat these simulated experiments below.
7.2.1 All or None Synaptic Induced Firing
The strength of an excitatory synapse may be modulated through
c
e
and was found to initiate an action
potential burst in an all-or-none fashion. Figure 7.2 shows Pyramidal cell 1 connected to Pyramidal cell 2
through an excitatory synapse at the soma of cell 2. A burst was initiated at the soma of cell 1 and the
strength of the synapse connection (
c
e
) was varied. It is clear from the figure that bursting in cell 2 is a
governed by the strength of the connection.
In large scale models, it is difficult to always draw the shape of every neuron, so many presentations
of
neural circuits
use some combination of circles to represent cells and triangles to represent synapses. In
Fig. 7.2, the excitatory connection between two Pyramidal cells is represented by two open circles (e.g.,
excitatory cells) connected together by a line terminating in an open triangle (excitatory synapse). The
direction of the triangle also indicates that cell 1 is driving cell 2.
