A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue


It is proposed that distinct anatomical regions of cerebral cortex and of thalamic nuclei are functionally two-dimensional. On this view, the third (radial) dimension of cortical and thalamic structures is associated with a redundancy of circuits and functions so that reliable signal processing obtains in the presence of noisy or ambiguous stimuli.

A mathematical model of simple cortical and thalamic nervous tissue is consequently developed, comprising two types of neurons (excitatory and inhibitory), homogeneously distributed in planar sheets, and interacting by way of recurrent lateral connexions. Following a discussion of certain anatomical and physiological restrictions on such interactions, numerical solutions of the relevant non-linear integro-differential equations are obtained. The results fall conveniently into three categories, each of which is postulated to correspond to a distinct type of tissue: sensory neo-cortex, archior prefrontal cortex, and thalamus.

The different categories of solution are referred to as dynamical modes. The mode appropriate to thalamus involves a variety of non-linear oscillatory phenomena. That appropriate to archior prefrontal cortex is defined by the existence of spatially inhomogeneous stable steady states which retain contour information about prior stimuli. Finally, the mode appropriate to sensory neo-cortex involves active transient responses. It is shown that this particular mode reproduces some of the phenomenology of visual psychophysics, including spatial modulation transfer function determinations, certain metacontrast effects, and the spatial hysteresis phenomenon found in stereopsis.

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Post-synaptic membrane potential (psp)

α :

Maximum amplitude of psp

t :


μ :

The neuronal membrane time constant

ϑ :

Threshold value of membrane potential

r :

Absolute refractory period

τ :

Synaptic operating delay

v :

Velocity of propagation of action potentil

x :

Cartesian coordinate

β jj′ (x):

The probability that cells of class j′ are connected with cells of class j a distance x away

b jj′ :

The mean synaptic weight of synapses of the jj-th class at x

σ jj′ :

The space constant for connectivity

ϱ e :

Surface density of excitatory neurons in a one-dimensional homogeneous and isotropic tissue

ϱ i :

Surface density of inhibitory neurons in a one-dimensional homogeneous and isotropic tissue

E(x, t):

Excitatory Activity, proportion of excitatory cells becoming active per unit time at the instant t, at the point x

I(x, t):

Inhibitory Activity, proportion of inhibitory cells becoming active per unit time at the instant t, at the point x

δx :

A small segment of tissue

δt :

A small interval of time

P(x, t):

Afferent excitation or inhibition to excitatory neurons

Q(x, t):

Afferent excitation or inhibition to inhibitory neurons

N e (x, t):

Mean integrated excitation generated within excitatory neurons at x

N i (x, t):

Mean integrated excitation generated within inhibitory neurons at x

e [N e ]:

Expected proportion of excitatory neurons receiving at least threshold excitation per unit time, as a function of N e

i [N i ]:

Expected proportion of inhibitory neurons receiving at least threshold excitation per unit time, as a function of N i

G(ρ e ):

Distribution function of excitatory neuronal thresholds

G(ρ 1 ):

Distribution function of inhibitory neuronal thresholds

ρ 1 :

A fixed value of neuronal threshold

h(N e ; ρ 1):

Proportion per unit time of excitatory neurons at x reaching ρ 1 with a mean excitation N e

1[ ]:

Heaviside's “step-function”

R e (x, t):

Number of excitatory neurons which are sensitive at the instant t

R i (x, t):

Number of inhibitory neurons which are sensitive at the instant t

R e :

Refractory period of excitatory neurons

r i :

Refractory period of inhibitory neurons

E(x, t)〉:

Time coarse-grained excitatory activity

I(x, t)〉:

Time coarse-grained inhibitory activity


Spatial convolution

θ :

Threshold of a neuronal aggregate

v :

Sensitivity coefficient of response of a neuronal aggregate


Time coarse-grained spatially localised excitatory activity

I(t)> :

Time coarse-grained spatially localised inhibitory activity

L 1,L 2,L,Q :

See § 2.2.1, § 2.2.7, § 3.1

υ :

Velocity with which retinal images are moved apart

σ :

Stimulus width

E o〉, 〈I o〉:

Spatially homogeneous steady states of neuronal activity

k e ,k ij S e S ij :

See § 5.1


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Wilson, H.R., Cowan, J.D. A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik 13, 55–80 (1973). https://doi.org/10.1007/BF00288786

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  • Prefrontal Cortex
  • Thalamic Nucleus
  • Modulation Transfer Function
  • Stable Steady State
  • Hysteresis Phenomenon