Authors: Warren S. McCulloch and Walter Pitts
Published in: Bulletin of Mathematical Biology, Vol. 52, No. 1/2, 1990, pp. 99-115.
Abstract
The paper presents a logical framework for understanding nervous activity through the lens of propositional logic, emphasizing the “all-or-none” characteristic of neural events. It explores how neural networks can be described logically, the equivalence of various neurophysiological assumptions, and the implications for theoretical neurophysiology.
1. Introduction
Theoretical neurophysiology is built on fundamental assumptions regarding the structure and function of the nervous system, which consists of neurons connected by synapses. Key points include:
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Each neuron has a threshold that must be exceeded to initiate an impulse.
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Impulses propagate along axons at varying velocities depending on the diameter of the axon.
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Synaptic transmission predominantly occurs from axon terminals to the soma of the next neuron.
The authors discuss the significance of excitatory and inhibitory synapses, the concept of temporal summation, and the complex interactions of neurons that lead to nervous impulses.
2. The Theory: Nets Without Circles
Physical Assumptions
The authors establish several key assumptions for their logical calculus:
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All-or-None Process: Neuron activity operates as an “all-or-none” process.
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Fixed Synaptic Excitation: A specific number of synapses must be activated within a certain timeframe to excite a neuron.
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Significant Delay: The only significant delay in the nervous system is synaptic delay.
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Inhibitory Synapse Activity: An active inhibitory synapse prevents excitation of the neuron.
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Static Structure: The net’s structure does not change over time.
Symbolism and Notation
The authors utilize a symbolic logic framework, incorporating elements from Carnap’s Language II and Russell and Whitehead’s Principia Mathematica. They define temporal propositional expressions (TPEs) and introduce a functor to handle properties of neurons.
Theorems
Several theorems are presented regarding the realizability of neural nets:
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Theorem 1: Every net of order 0 can be expressed in terms of TPEs.
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Theorem 2: Every TPE is realizable by a net of order zero.
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Theorem 3: Complex sentences built from elementary propositions are TPEs under specific conditions.
3. The Theory: Nets With Circles
Challenges with Cyclic Nets
The authors address the increased complexity of analyzing nets containing circles, which can exhibit indefinite patterns of activity. The paper describes methods for expressing the actions of neurons within cyclic networks and emphasizes the need for iterative substitution to solve equations.
Realizability
The authors discuss the conditions necessary for realizability in cyclic nets, establishing that certain neural conditions allow for the derivation of past states from present activity. They introduce the concept of “prehensible classes,” which can be defined using logical operations.
4. Consequences
Causality and Knowledge
The authors explore the implications of their logical framework for understanding causality in neuroscience. They argue that while present activities can predict future states, deducing past activities remains uncertain due to the nature of disjunctions and the regenerative activity of cyclic nets. This leads to insights about the inherent incompleteness of knowledge regarding the world.
Application to Psychology and Neurology
The framework provides a basis for understanding mental processes in terms of neural activities. The distinction between equivalent nets and their implications for psychological phenomena is highlighted, along with the potential for a clearer understanding of neurophysiological disorders.
Conclusion
The paper concludes by asserting the relevance of their logical calculus in bridging the gap between neurophysiology and psychological phenomena, underscoring its utility in theoretical studies and practical applications within the fields of psychology and neurology.
References
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Carnap, R. (1938). The Logical Syntax of Language. New York: Harcourt-Brace.
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Hilbert, D., & Ackermann, W. (1927). Grundtige der Theoretischen Logik. Berlin: Springer.
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Russell, B., & Whitehead, A. N. (1925). Principia Mathematica. Cambridge University Press.
This summary encapsulates the key themes and findings of McCulloch and Pitts’ work, providing a structured overview of their contributions to the understanding of nervous activity through logical analysis.