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Comparison of models for subtractive and shunting lateral-inhibition in receptor-neuron fields

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Summary

Two types of neuronal lateral inhibition in one-dimensional fields of receptors and neurons are considered. The first type, which has been demonstrated in the eye of Limulus, is called subtractive inhibition (SI): it assumes that neuronal activity depends on the difference between the total excitation and inhibition. The second type is called shunting inhibition (SHI): it assumes that inhibitory influences cause a shunting of a portion of the excitation-produced depolarizing current. Consideration of the shunting model is dictated by its considerable physiological plausibility. The actions of SI and SHI, examined for a variety of coupling conditions and time-stationary positive inputs, are shown to be markedly different. The results indicate that SI is most suited for obtaining (1) a linearity between input and output, (2) a contrasting effect that does not depend on the presence of input discontinuities, and (3) contrasting whose degree is independent of input amplitude. SI is especially useful if coupling coefficients can be varied to accommodate the various input form functions or if, for fixed coupling coefficients, the class of input form functions is limited. On the other hand SHI appears most suited for obtaining (1) a nonlinear input-output relation, (2) a relative contrasting only of discontinuities, and (3) a dependence of the contrasting upon input amplitude.

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Abbreviations

a :

coupling coefficient for neighboring units, also called coupling amplitude

V j :

output of receptor number j

i j :

generator current of neuron number j

g :

inhibitory function for subtractive inhibition

h :

inhibitory function for shunting inhibition

ϱ :

v 2/v 1 [applies to two-unit case]

N k :

neuron number k

I k :

total source current produced by excitatory influences on N k

G k :

conductance for source current not shunted (with shunting inhibition)

i σ :

portion of source current shunted as a result of inhibition

m :

number of inhibitory influences [in Eq. (1)]

G kj :

conductance of inhibitory shunt path j for neuron N k

q :

number of receptors

n :

number of neurons

R j :

receptor number j

x :

distance

y(x) :

input stimulus to receptors

y j =y(x j ):

input stimulus to receptor R j

v j :

vj for v j ≧0, zero otherwise

a kj :

G kj /v j , inhibitory coupling coefficient for forward shunting inhibition [refer to Eq. (2)]

b kj :

excitatory coupling coefficient for contribution to source current of neuron N k by receptor R j [refer to Eq. (3)]

i j :

i j for i j ≧0, zero otherwise

c kj :

G kj /i j , inhibitory coupling coefficient for backward shunting inhibition [refer to Eq. (4)]

â kj :

inhibitory coupling coefficient for forward subtractive inhibition [refer to Eq. (5)]

ĉ kj :

inhibitory coupling coefficient for backward subtractive inhibition [refer to Eq. (6)]

y(x j )=Af(x j ):

sensory input function

A :

input amplitude

f(x j ):

sensory input form function, also called a sensory image

i(x j ):

generator current output of neuron Nj which is located at x=x j

y :

(y 1, y 2, ..., y n), a column vector

i :

(i 1, i 2, ..., i n), a column vector, also called generator current configuration

a :

an n by n matrix having a kj as the term in the k-th row, j-th column

U :

the unit matrix

d :

¦k-j¦, separation between neurons N k and N j

a :

a kj for d=1, called coupling amplitude

SI:

subtractive inhibition

SHI:

shunting inhibition

FSI:

forward subtractive inhibition

BSI:

backward subtractive inhibition

FSHI:

forward shunting inhibition

BSHI:

backward shunting inhibition

s :

i/i 51 = (s 1, s 2, ..., s n), normalized generator current vector, also called normalized generator current configuration

s j :

i j/i 51, normalized generator current of neuron N j

f(x) :

continuous input form function of which f(x j ) is a sampled version

pf(x)/∂xp :

p-th order derivative of f(x)

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Furman, G.G. Comparison of models for subtractive and shunting lateral-inhibition in receptor-neuron fields. Kybernetik 2, 257–274 (1965). https://doi.org/10.1007/BF00274089

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Keywords

  • Neuronal Activity
  • Lateral Inhibition
  • Contrasting Effect
  • Coupling Coefficient
  • Form Function