Editing Warren Sturgis McCulloch (section)

== Work ==
He is remembered for his work with Joannes Gregorius Dusser de Barenne from Yale<ref>{{cite journal | pmc = 2602446 | volume=12 | issue=6 | journal=Yale J Biol Med | title=Joannes Gregorius Dusser de Barenne| pages=742.2–746 | year=1940 | pmid=21433922 }}</ref> and later with [[Walter Pitts]] from the [[University of Chicago]]. He provided the foundation for certain brain theories in a number of classic papers, including "''[[A Logical Calculus of the Ideas Immanent in Nervous Activity]]''" (1943) and "''How We Know Universals:  The Perception of Auditory and Visual Forms''" (1947), both published in the ''Bulletin of Mathematical Biophysics''. The former is "widely credited with being a seminal contribution to neural network theory, the theory of automata, the theory of computation, and cybernetics".<ref name = "KA04"/>

McCulloch was the chair of the set of [[Macy conferences]] dedicated to Cybernetics. These, greatly due to the diversity of the backgrounds of the participants McCulloch brought in, became the foundation for the field.

In [[Cybernetics: Or Control and Communication in the Animal and the Machine|Wiener's ''Cybernetics'']] (1948), he recounted an event in the spring of 1947, when McCulloch designed a machine to allow the blind to read, by converting printed letters to tones. He designed it so that the tone is invariant for the same letter viewed under different angles. Gerhardt von Bonin saw the design, and immediately asked, " Is this a diagram of the fourth layer of the [[visual cortex]] of the brain?".<ref>Wiener, Norbert. ''Cybernetics or Control and Communication in the Animal and the Machine''. (1948)</ref>{{Pg|pages=22, 140}}

In his last days in 1960s, he worked on loops, oscillations and triadic relations with Moreno-Díaz; the reticular formation with Kilmer and dynamic models of memory with Da Fonseca.<ref>{{Citation |last1=de Blasio |first1=Gabriel |title=McCulloch's Relation to Connectionism and Artificial Intelligence |date=2018 |work=Computer Aided Systems Theory – EUROCAST 2017 |volume=10671 |pages=41–48 |editor-last=Moreno-Díaz |editor-first=Roberto |url=http://link.springer.com/10.1007/978-3-319-74718-7_6 |access-date=2024-10-12 |place=Cham |publisher=Springer International Publishing |doi=10.1007/978-3-319-74718-7_6 |isbn=978-3-319-74717-0 |last2=Moreno-Díaz |first2=Arminda |last3=Moreno-Díaz |first3=Roberto |editor2-last=Pichler |editor2-first=Franz |editor3-last=Quesada-Arencibia |editor3-first=Alexis}}</ref> His work in the 1960s was summarized in a 1968 paper.<ref name=":0">{{Citation |last=McCulloch |first=W. S. |title=Logic and Closed Loops for a Computer Junket to Mars |date=1968 |work=Neural Networks |pages=65–91 |editor-last=Caianiello |editor-first=E. R. |url=http://link.springer.com/10.1007/978-3-642-87596-0_7 |access-date=2024-10-12 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |language=en |doi=10.1007/978-3-642-87596-0_7 |isbn=978-3-642-87598-4}}</ref>

=== Neuroscience ===
He studied the excitation of the brain by [[strychnine]] neuronography, which was a method to map brain connections. Applying strychnine in one point of the brain causes excitations in different points of the brain.<ref>{{Cite journal |last1=de Barenne |first1=J. G. Dusser |last2=McCulloch |first2=W. S. |date=1939-10-31 |title=Physiological Delimitation of Neurones in the Central Nervous System |url=https://www.physiology.org/doi/10.1152/ajplegacy.1939.127.4.620 |journal=American Journal of Physiology. Legacy Content |language=en |volume=127 |issue=4 |pages=620–628 |doi=10.1152/ajplegacy.1939.127.4.620 |issn=0002-9513}}</ref> Bailey, Bonin, and McCulloch conducted a series of studies in the 1940s that identified connections in the brains of macaque and chimpanzee that are consistent with modern understanding of [[Vertical occipital fasciculus|VOF]].<ref>''The isocortex of the chimpanzee.'' by Percival Bailey, Gerhardt von Bonin, and Warren S. McCulloch. Urbana, Illinois, University of Illinois Press, 1950</ref><ref>{{Cite journal |last1=Takemura |first1=Hiromasa |last2=Pestilli |first2=Franco |last3=Weiner |first3=Kevin S. |date=2019-09-01 |title=Comparative neuroanatomy: Integrating classic and modern methods to understand association fibers connecting dorsal and ventral visual cortex |journal=Neuroscience Research |volume=146 |pages=1–12 |doi=10.1016/j.neures.2018.10.011 |pmid=30389574 |pmc=6491271 |issn=0168-0102}}</ref>

=== Mathematical logic ===
In 1919 he began to work mainly on mathematical logic, and by 1923 he attempted to make a logic of [[transitive verb]]s. His goal in psychology is to invent a "psychon" or "least psychic event" that are binary atomic events with necessary causes, such that they can be combined to create complex logical propositions concerning their antecedents. He noticed in 1929 that these may correspond to the all-or-nothing firings of neurons in the brain.<ref name="number" />

In the 1943 paper, they described how memories can be formed by a neural network with loops in it, or alterable synapses. These then encodes for sentences like "There was some x such that x was a ψ" or <math>(\exists x) (\psi x)</math>, and showed that looped neural networks can encode all [[First-order logic|first-order logic with equality]] and conversely, any looped neural networks is equivalent to a sentence in first-order logic with equality, thus showing that they are equivalent in logical expressiveness.<ref name="number" />

The 1943 paper describes neural networks operating over time, and [[Universal quantification|logical universals]] -- "there exists" and "for all"—for spatial objects, such as geometric figures, was further developed in their 1947 paper.<ref>{{Cite journal |last1=Pitts |first1=Walter |last2=McCulloch |first2=Warren S. |date=1947-09-01 |title=How we know universals the perception of auditory and visual forms |url=https://link.springer.com/article/10.1007/bf02478291 |journal=The Bulletin of Mathematical Biophysics |language=en |volume=9 |issue=3 |pages=127–147 |doi=10.1007/BF02478291 |pmid=20262674 |issn=1522-9602}}</ref>

He worked with [[Manuel Blum]] in studying how a neural network can be "logically stable", that is, can implement a boolean function even if the activation thresholds of individual neurons are varied.<ref>Blum, Manuel. "Properties of a neuron with many inputs." ''Bionics Symposium: Living Prototypes--the Key to New Technology, 13-14-15 September 1960''. WADD technical report, 60-600. (1961)</ref>{{Pg|page=64}} They were inspired by the problem of how the brain can perform the same functions, such as breathing, under influence of [[caffeine]] or [[Ethanol|alcohol]], which shifts the activation threshold over the entire brain.<ref name="number" />

He worked on triadic relations, an extension of the [[Algebraic logic|calculus of relations]] to handle relations that relates 3 objects, such as "A gives B to C" or "A perceives B to be C". He was convinced that such a logic is necessary for understanding brain activity.<ref name=":0" /><ref name=":1">{{Cite journal |last=Arbib |first=Michael A |date=2000 |title=Warren McCulloch's Search for the Logic of the Nervous System |url=https://muse.jhu.edu/pub/1/article/46496 |journal=Perspectives in Biology and Medicine |volume=43 |issue=2 |pages=193–216 |doi=10.1353/pbm.2000.0001 |issn=1529-8795}}</ref>

=== How we know universals ===
In the 1947 paper ''How we know universals'', they studied the problem of recognizing objects despite changes in representation. For example, recognizing a square under different viewing angles and lighting conditions, or recognizing a phoneme under different loudness and tones. That is, recognizing objects invariant under the [[Group action|action]] of some [[symmetry group]]. This problem was partly inspired by a practical problem in designing a machine for the blind to read (recounted in Wiener's ''Cybernetics'', see before).<ref>{{Citation |last=Masani |first=P. R. |title=McCulloch, Pitts and the Evolution of Wiener's Neurophysiological Ideas |date=1990 |work=Norbert Wiener 1894–1964 |pages=218–238 |url=http://link.springer.com/10.1007/978-3-0348-9252-0_16 |access-date=2024-10-13 |place=Basel |publisher=Birkhäuser Basel |language=en |doi=10.1007/978-3-0348-9252-0_16 |isbn=978-3-0348-9963-5}}</ref>

The paper proposed two solutions. The first is in computing an invariant by averaging over the symmetry group. Let the symmetry group be <math>G</math> and the object to be recognized be <math>x</math>. Let a neural network implement a function <math>T</math>. Then, the group-invariant representation would be <math>\frac{1}{|G|}\sum_{g \in G} T(g x)</math>, the group-action average. The second solution is in a negative feedback circuit that drives a canonical representation. Consider the problem of recognizing whether an object is a square. The circuit moves the eye so that the "center of gravity of brightness" of the object is moved to the middle of the visual field. This then effectively converts each object into a canonical representation, which can then be compared with a representation in the brain.<ref>{{Cite journal |last=Aizawa |first=Kenneth |date=September 2012 |title=Warren McCulloch's Turn to Cybernetics: What Walter Pitts Contributed |url=https://journals.sagepub.com/doi/full/10.1179/0308018812Z.00000000017 |journal=Interdisciplinary Science Reviews |language=en |volume=37 |issue=3 |pages=206–217 |doi=10.1179/0308018812Z.00000000017 |bibcode=2012ISRv...37..206A |issn=0308-0188}}</ref><ref>{{Citation |last=Masani |first=P. R. |title=McCulloch, Pitts and the Evolution of Wiener's Neurophysiological Ideas |date=1990 |work=Norbert Wiener 1894–1964 |pages=218–238 |url=http://link.springer.com/10.1007/978-3-0348-9252-0_16 |access-date=2024-10-14 |place=Basel |publisher=Birkhäuser Basel |language=en |doi=10.1007/978-3-0348-9252-0_16 |isbn=978-3-0348-9963-5}}</ref>

=== Neural network modelling ===
In the 1943 paper McCulloch and Pitts attempted to demonstrate that a [[Turing machine]] program could be implemented in a finite network of ''formal'' [[neuron]]s (in the event, the Turing Machine contains their model of the brain, but the converse is not true<ref>see: S.C. Kleene, "Representations of Events in Nerve Nets and Finite Automata"</ref>), that the neuron was the base logic unit of the brain. In the 1947 paper they offered approaches to designing "nervous nets" to recognize visual inputs despite changes in orientation or size.

From 1952 McCulloch worked at the Research Laboratory of Electronics at MIT, working primarily on [[neural network (biology)|neural network]] modelling. His team examined the visual system of the [[frog]] in consideration of McCulloch's 1947 paper, discovering that the eye provides the brain with information that is already, to a degree, organized and interpreted, instead of simply transmitting an image.

With Roberto Moreno-Díaz, he studied a formalized problem of memory. Given that neural networks can story memory by a pattern of oscillations in a circle, they studied the number of possible oscillation patterns that can be sustained by some neural network with <math>N</math> neurons. This came out to be <math>K(N) = \binom{2^N}{k}\sum_{k=1}^{2^N-1} k!</math> (Schnabel, 1966).<ref>Schnabel, C. P. J. ''Number of Modes of Oscillation of a Net of N Neurons''. Quarterly Progress Report No. 80, Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Mass., January 15, 1966. P. 253.</ref> Also, they proved a universality theorem, in that for each <math>N</math>, there exists a neural network (possibly with more than <math>N</math> neurons) with <math>\log_2 K(N)</math> binary inputs, such that, for any oscillation pattern realizable by some neural network with <math>N</math> neurons, there exists a binary input for this universal network such that it exhibits the same pattern.<ref>Moreno-Díaz, R. ''Realizability of a Neural Network Capable of All Possible Modes of Oscillation''. Quarterly Progress Report No. 82, Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Mass., July 15, 1966. Pp. 280-285.</ref><ref>Moreno-Díaz, R., and W. S. McCulloch. "''Circularities in nets and the concept of functional matrices''." ''Biocibernetics of the Central Nervous System, Little, Brown and Company, Boston'' (1969): 145-151.</ref><ref>{{Cite journal |last1=Moreno-Díaz |first1=Roberto |last2=Moreno-Díaz |first2=Arminda |date=2007-04-01 |title=On the legacy of W.S. McCulloch |url=https://www.sciencedirect.com/science/article/abs/pii/S0303264706002152 |journal=Biosystems |series=BIOCOMP 2005: Selected papers presented at the International Conference - Diffusion Processes in Neurobiology and Subcellular Biology |volume=88 |issue=3 |pages=185–190 |doi=10.1016/j.biosystems.2006.08.010 |bibcode=2007BiSys..88..185M |issn=0303-2647}}</ref>

=== Control ===
McCulloch considered the problem of contradictory information and motives, which he called a "[[heterarchy]]" of motives, meaning that the motives are not linearly ordered, but can be ordered like <math>A > B > C > A</math>.<ref>{{Cite journal |last=McCulloch |first=Warren S. |date=June 1945 |title=A heterarchy of values determined by the topology of nervous nets |url=http://link.springer.com/10.1007/BF02478457 |journal=The Bulletin of Mathematical Biophysics |language=en |volume=7 |issue=2 |pages=89–93 |doi=10.1007/BF02478457 |pmid=21006853 |issn=0007-4985}}</ref> He posited the concept of "poker chip" [[reticular formation]]s as to how the brain deals with contradictory information in a democratic, somatotopical neural network. Specifically, how the brain can commit the animal to a single course of action when the situation is ambiguous.

They designed a prototypic example neural network "RETIC", with "12 anastomatically coupled modules stacked in columnar array", which can switch between unambiguous stable modes based on ambiguous inputs.<ref name=":2" /><ref name=":1" />

His principle of  "Redundancy of Potential Command"<ref name=":2">{{Citation |last1=Kilmer |first1=W. L. |title=Some Mechanisms for a Theory of the Reticular Formation |date=1968 |work=Systems Theory and Biology |pages=286–375 |editor-last=Mesarović |editor-first=M. D. |url=http://link.springer.com/10.1007/978-3-642-88343-9_14 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |language=en |doi=10.1007/978-3-642-88343-9_14 |isbn=978-3-642-88345-3 |last2=McCulloch |first2=W. S. |last3=Blum |first3=J.}}</ref> was developed by [[Heinz von Foerster|von Foerster]] and [[Gordon Pask|Pask]] in their study of [[self-organization]]<ref>"A Predictive Model for Self-Organizing Systems", Part I: Cybernetica 3, pp. 258–300; Part II: Cybernetica 4, pp. 20–55, 1961 [[Heinz von Foerster]] and [[Gordon Pask]]</ref> and by Pask in his [[Conversation Theory]] and [[Gordon Pask#Interactions of Actors Theory|Interactions of Actors Theory]].<ref name="Pask 1996">Gordon Pask (1996). ''Heinz von Foerster's Self-Organisation, the Progenitor of Conversation and Interaction Theories''</ref>