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and chemical synaptic currents, V
V
1
V
N
is an N-dimensional vector
comprised of the voltages of all N neurons in the network, and I
sen
i
t represents
chemosensory input (from Ferrée & Lockery, p. 14).
They then borrow from data on
Ascaris
, since as frequently noted in the worm
literature, synaptic neurophysiological data are not yet available for
C. elegans
,
so a closely related worm,
Ascaris
, is used. This data allows them to assert that
the chemical synapses between cells i and j can be modeled by the a sigmoidal
functional equation
=
j
=
1
N
ij
V
j
−
V
j
I
chem
G
chem
V
=
·
·
V
i
−
E
ij
(2)
i
ij
where G
chem
ij
is the maximum conductance in the cell i because of synaptic
connections from cell j and E
ij
is the reversal potential for the correspond-
ing postsynaptic current. Electrical synapses are similarly modeled by another
slightly simpler third equation. Further, chemical inputs to the system are cap-
tured by
I
sen
i
t
=−
il
sens
Ct
(3)
where Ct is the chemical concentration at the tip of the worm's nose,
i1
is
the standard Kronecker delta and
sens
is a constant parameter.
The total synaptic model can be further simplified by representing only the
chemical synapses. Equation (2), which is then governing, is nonlinear, but it
can be linearized by using a Taylor series expansion (familiar to elementary
calculus students) and retaining only the linear terms. This process yields the
following set of equations:
j
=
1
=
N
dV
i
dt
=
A
ij
V
j
+
b
i
+
c
i
t
(4)
(The matrix A
ij
and b
j
are complicated functions of the G
s, V
s, and E
s intro-
duced in Eqns (1) and (2), and are not reproduced here; see Ferrée & Lockery,
1999, pp. 16-17.) This linearized equation and two quite simple body model
equations are then combined with an equation representing the chemical envi-
ronment, C, and the equations solved to yield a state trajectory St that begins
from some specified initial state S
0
. The simulation solutions were obtained
by numerical integration, akin to H and H's work, though now using power-
ful computer tools, and some other tricks employed to eliminate transients. In
their paper, Ferrée and Lockery show a comparison between real and simulated
