Mathematical Psychology and Psychophysiology

By Stephen Grossberg | Go to book overview

so that

. From this it follows this it follows that sτρ can take values between 0 and 1. Similarity between two retinal fibres is measured in an identical fashion.

The concentrations in a given tectal cell τ are a mixture of those induced by the retinal fibres and those supplied by lateral transport within the tectum. A desired property of the system is that fibres should concentrate their contacts in compact regions of the tectum, and that tectum cells should receive contacts from compact regions of the retina. The first is ensured by the competition that is a consequence of the sum rule (2); the second is the interference realized by calculating the similarities as a function of the normalized concentrations. The calculation of similarity in this way has a plausible physiological basis; for more details see [2].

A final remark, regarding the form of differential equation (3), may be added here. Grossberg [7] discusses the differential equation:

, (10)
in which xi is the degree of excitation of cell i in a field of cells, B an upper bound to xi, α a decay rate constant and Ii an input impinging on cell i. Grossberg refers to the last two terms as an on-center off-surround shunting interaction. The equation has the pleasing property that in the stationary case of , i.e. the ratio of two xi's depends on the ratios of the corresponding Ii's, irrespective of the total input . In the special case of α = 0, (10) becomes:
, (11)
which has the same form as (3). This shows that the special case of (10) with α = 0 can be derived from the 'rules' ẋi = BIi and in the same way as (3) was derived from (1) and (2). The case of α ≠ 0 cannot be introduced into the differential equation in this way, as modification of the first 'rule' to i = -αxi + BIi still leads to the same equation (11). It may be regarded as a very interesting fact that the three differential equations, (3), (10), and equation (37) of [6], although having very different meaning and interpretation, bear so close formal relationships.

ACKNOWLEDGEMENT. We would like to thank Dr. A. Häussler, who proposed important improvements to the manuscript.


REFERENCES

1. R. M. Gaze, The problem of specificity in the formation of nerve connections, Specificity of Embryological Interactions ( D. Garrod, ed.), Chapman and Hall, London, 1978.

2. D. J. Willshaw and C. von der Malsburg, A marker induction mechanism for the establishment of ordered neural mappings: its application to the retinotectal problem, Philos. Trans. Roy. Soc. London Ser. B 287 ( 1979), 203-243.

3. C. von der Malsburg and D. J. Willshaw, How to label nerve cells so that they can interconnect in an ordered fashion, Proc. Nat. Acad. Sci. U.S.A. 74 ( 1977), 5176-5178.

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