A computational model of vertical signal

Biological
Cy metics

Biol. Cybern. 68, 43-52 (1992)

9 Springer-Verlag 1992

A computational model of vertical signal propagation
in the primary visual cortex *
Paul Patton ~, Elizabeth Thomas 2, and Robert E. Wyatt 2
i Arizona Research Laboratories, Division of Neural Systems, Memory, and Aging, 384 Life Sciences North Building, Tucson, AZ 85724, USA
2 Department of Chemistry and Institute of Theoretical Chemistry, University of Texas, Austin, TX 78712, USA
Received August 15, 1990/Accepted in revised form May 19, 1992

Abstract. A computational model of the flow of activ-

ity in a vertically organized slab of cat primary visual
cortex (area 17) has been developed. The membrane
potential of each cell in the model, as a function of
time, is given by the solution of a system of first order,
coupled, non-linear differential equations. When firing

threshold is exceeded, an action potential waveform is
"pasted" in. The behavior of the model following a
brief simulated stimulus to afferents from the dorsal
lateral geniculate nucleus (dLGN) is explored. Excitatory and inhibitory post-synaptic potential (E and
IPSP) latencies, as a function of cortical depth, were
generated by the model. These data were compared
with the experimental literature. In general, good agreement was found for EPSPs. Many disynaptic inhibitory
inputs were found to be "masked" by the firing of
action potentials in the model. To our knowledge this
phenomenon has not been reported in the experimental
literature. The model demonstrates that whether a cell
exhibits disynaptic or polysynaptic PSP latencies is not
a fixed consequence of anatomical connectivity, but
rather, can be influenced by connection strengths, and
may be influenced by the ongoing pattern of activity in
the cortex.

1 Introduction

This study concerns the development of a model of a

small patch of mammalian primary visual cortex with
the goal being the simulation of the dynamics of cortical activity flow. The model incorporates a great deal of
anatomical and physiological detail. It is intended to
simulate the vertical flow of activity in a slab of cortical
tissue that extends from the pial surface of the brain to
the subcortical white matter and laterally over a
500 x 500 ~tm2 patch of cortical surface. The behavior

* Supported by a grant from Cray Research Inc.
Correspondence to: R. E. Wyatt

of such a slab following the delivery of a brief stimulus
to cortical afferents from the dorsal lateral geniculate
nucleus (dLGN) is computed. The results will be compared with experimental studies of activity flow and
signal latencies in cat primary visual cortex.
A number of investigators have attempted to model
activity flow in the cerebral cortex. Wilson and Cowan
(1973) considered only the horizontal organization of
the cortex. Douglas et al. (1989) have proposed a
canonical circuit to describe the interaction between

excitatory and inhibitory neurons in cortex. They did
not consider individual neurons, but modeled only population interactions. Gremillion et al. (1987) have presented only preliminary results. Recent experimental
findings (Gray and Singer 1988; Gray et al. 1989) have
shown that neurons in cat primary visual cortex can
exhibit oscillatory responses that are coupled over long
distances and sensitive to global stimulus properties.
This has prompted a flurry of computational investigations of these oscillations. The study of Krone et al.
(1986) is, to our knowledge, the only previous computational study to consider the vertical flow of activity in
cortex. However, they made no attempt to compare the
timecourse of laminar spread of activity in their model
with experimental results.
The anatomical properties of the model have been
described in a previous report (Thomas et al. 1991).
Here we will first report the construction of the physiological component of the model. We will then describe
the timecourse of vertical spread of activity in the
computational model following a brief synchronous
stimulus to the thalamic afferents, such as might be
produced experimentally by an electrical stimulus to the
optic radiation. A videotape, titled Brain Glow, has
been produced to display these results (Wyatt and

Driver 1991). This behavior will be compared with
experimental reports of activity and signal latencies in
cat visual cortex following the delivery of such stimuli.
The model will be used to explore the relationship
between post-synaptic potential latencies and the ordinal positions of cells with respect to the thalamic afferents. The results will be used to assess the plausibility of

44
drawing conclusions about ordinal position from measurements of signal latency.

2 Constructing the model physiology
2.1 An overview o f the model physiology

The physiological properties of the model, as introduced
here, are based on physiological data reported for rodents
as well as for the cat. To permit the computation of
the membrane potential for each neuron in a large
network over time, each cell is treated as a single
physiological compartment. Membrane potential in each
compartment is given by a first order, nonlinear differential equation. The coupled system of equations is then
solved numerically.

Both pyramidal and spinous stellate neurons in the
model are assigned excitatory properties (Douglas and
Martin 1990; Chagnac-Amitai and Connors 1989). The
same was done for d L G N afferents (Ferster and Linstrom
1983; Bullier and Henry 1979). All excitatory conductances used in the current model are based on the
timescale and magnitude of n o n - N M D A mediated conductances. In this model, we deal only with the behavior
of the model on a relatively short time scale and have
therefore excluded the N M D A conductance, which acts
on a long timescale (MacDermott and Dale 1987). The
remaining cell classes, which together constitute the
smooth neurons (Peters and Proskauer 1980; Somogyi
and Cowey 1981; Freund et al. 1983; Morrison et al. 1984;
Peters and Kimerer 1981) were represented as inhibitory
neurons. GABAA receptors appear to be localized in the
soma and G A B A s receptors in the dendritic arbor
(Connors et al. 1988). This feature was incorporated in
the model. As a consequence of this treatment, and the
connection scheme chosen, chandelier cells in the model
make only G A B A A connections. In addition to synaptic
conductances, each model neuron is endowed with an

afterhyperpolarization and leak conductance.
Three distinct physiological classes of neurons have
so far been identified in mammalian neocortex: regular
spiking (RS), fast spiking (FS), and intrinsic burst (IB)
neurons (McCormick et al. 1985; Connors and Gutnick
1990). They differ from one another in terms of spike
duration, firing rates, adaptation, and patterns of afterhyperpolarization and afterdepolarization. Intrinsic
burst neurons fire all or none bursts of 3 to 5 action
potentials. Most pyramidal neurons in the model are
assigned as RS cells, as are most spinous stellates. The
remainder, consisting of 10% of the pyramidal and
stellate neurons in layers 4 and 5, are assigned as IB
neurons. All inhibitory cell classes are treated as FS
neurons. In the model, the three physiological classes
were distinguished from one another only by spike
durations and firing rates (as reflected in absolute and
relative refractory period timecourses).
2.2 Mathematical characterization o f the model
2.2.1 Equations governing the behavior o f model neurons
in the subthreshold regime. Each neuron j is represented


as a single physiological compartment. A connection
between presynaptic cell i and postsynaptic cell j, in the
model, may represent one or several synapses. A coupling is the set of all connections between i and j. The
model "lumps" the connections impinging on cell j
from cell i into two groups: those that impinge on the
soma, and those that impinge on the dendritic arbor.
The strength of these two lumped connections is determined by the number of individual connections each
incorporates, as well as by its classification as somatic
or dendritic. The response of cell j to synaptic input is
governed by a first order non-linear differential equation as given below
C dVj(t)dt = I~eXS+.jt~.xa + ig,baa + i IgabaB ,3r _jlleak+i~hp

(1)
where Vj (t) is the membrane potential of cell j, C is the
membrane capacitance, i?xs, I~eXa,11gabaA, and 1IgabaB, are
synaptic currents, and I~*ak and I~hp are the leak and
afterhyperpolarization currents respectively. I~x* and
I~xd are the currents produced by all excitatory input
onto the soma and dendrites respectively. I f f b~A is the

current produced by all inhibitory inputs to cell j involving GABAa receptors and Iff b"B is the current
involving GABAs receptors. To compute the magnitude of any synaptic current I 7 n, we must first compute
the sum S~yn of the currents associated with each coupling of type syn:
P

S~yn = fl E N,~"ynOtijgSyn(t -- ta)(Vj(t) -- g syn)

(2)

i=1

In the equation, fl is a scaling factor that increases the
strength of each connection to compensate for the
reduced number of cells in the model. Since there is a
linear relationship between the number of cells in the
model and the number of connections per cell (Thomas
et al. 1991), fl is the ratio between the realistic number
of cells in a slab (roughly 19600) and the reduced
number of cells used in the simulation. Typical simulation runs are at 5% realistic cell density, -N~.
- U yn is the

number of connections of type syn in each coupling. ~,.j
is a term which adds a degree of randomness to the
strength of coupling. It is given by: ~ = 1 + (~xr~"),
where X ra" is a random number between - 1 and 1, and
c~is a constant which will be set equal to 0.5. E syn is the
reversal potential associated with the current syn.
gsyn(t- tij ) is a function giving the membrane conductance. In this expression, t is the current time and
t~ = t i + Z ~ , where t~ is the time when presynaptic cell i
fired an action potential, and T,~ is the mean transmission delay for axosomatic (n = soma) or axodendritic
(n = den) connections from cell i to j as discussed
below. This function is given by:
gSy"(t - to.) = K~Y~(t - tq)exp( - ( t -- tij)/T syn)

(3)

where g syn is a constant whose value can be computed
by replacing (t - t,j) in (3) with r~Yn; the time of occurrance of the peak conductance associated with this
current. The conductance in this case would be the peak
conductance


G syn

45
Given sjYn , synaptic current I~yn can be computed:
i~Yn = Fsyn sgn(Sjyn)gcut( 1 -- exp( - I S ~ yn I/gcut))

(4)

When F syn = 1, the equation prevents I~y~ from assuming
a value larger than cutoff K cut. The value F syn is used to
adjust the magnitude of I~y" to allow for the different
weights assigned to somatic versus dendritic connections.
The afterhyperpolarizing conductance g a h P ( t - tj),
(where tj is the time of firing of cell j), like the synaptic
conductances, is represented by the alpha function of
(3). The ahp current generated by this conductance is
given b y :
i~hP = g~hP(t _ tj)(Vj (t) -- E ahp)

(5)


w h e r e E ahp is the reversal potential for the ahp conduc-

tance. The magnitude of the leak current I ~e~k, is obtained by using g~e~k and E leak in (5).
Conduction distance was not computed as the most
direct route from a cell soma to the point of connection. It was, rather, the distance along lines parallel to
the coordinate axes, one must travel to go from soma
to connection. This was done to represent the often
indirect route taken by an axon in reaching its target.
The system of coupled first order non-linear differential equations describing the network was solved numerically using the forward Euler method on a Stardent
2000 graphics minisupercomputer. A time step of
0.01 ms was used in most simulations. Appropriate tests
were carried out to ensure the stability and consistency
of the integrator. When any neuron exceeds its firing
threshold, an action potential waveform is "pasted"
into the model as described below.

2.2.2 Equations governing the behavior o f model neurons
in the suprathreshold regime. Variation in membrane
potential for a neuron firing an action potential was
modeled as the first half period of a sine wave. The
firing pattern of IB neurons is simulated by generating
three half sine wave action potentials. The time between
these action potentials is determined by the absolute
refractory period of the cell.
Each cell is assigned an absolute and relative refractory period following the firing of an action potential.
During its absolute refractory period, a cell is unable to
fire. During the relative refractory period, firing threshold is elevated. McCormick et al. (1985) have studied
the relationship between the size of an injected current
and the firing rate of cortical neurons in response to the
injection. They have found that, for small current

T a b l e I.

injections, the relationship is linear; and that the slope
of the relationship for regular spiking neurons and for
fast spiking neurons is different. We have used this
relationship as the basis for assigning relative refractory
period. During the relative refractory period, the elevated firing threshold is assigned as

vY'~

__

J

-

'oF-

tj)

+

V thres

J

where

T; bs < t -- tj