Some thoughts about neural coding and spike trains
Jose´ Pedro Segundo *
,1Departamento de Biomatema´tica,Facultad de Clenclas,Monte6ideo,Uruguay
Abstract
This communication introduces the topic.Foundations:Core concepts: Codings are relations summarized by rules or ‘codes’. Special codings are ‘neural’, ‘natural’ (in everyday life), ‘experimental’ (in laboratories), ‘conditional’ (to partner restrictions), etc. Partial aspects are mechanisms, what partners say about each other, etc.Critical experimen -tal issues:Trains are e6aluated by when spikes occur: i.e.as point processes and timings. Trains and point process
representations become synonyms. Any code must: (i) be a ‘number (rate) cod’ and an ‘interval cod’; and (ii) include ‘referent, train’ covariations involving steady states with overall averages and fluctuations with patterns (dispersions, sequences). Seminal findings. Early data proved trains participated in codings; this is accepted unanimously. Inevitably, though accepted less readily, codings included rates, intervals, averages and patterns.Literature highlights. (1) Confirmed the seminal finding (2.2.) over vast domains; (2) Demonstrated both general and synaptic codings (referents, respectively, sensory, states, etc. and trains in directly connected neurons); (3) Revealed overlap between general and synaptic coding features. Overlap allows train participation in network dynamics; (4) Introduced natural formal contexts. (Point Process Mathematics, Communication. Information and Dynamical Systems Theories); (5) Includes confused opinions: (i) Opposition between rates and intervals; (ii) claims that averages are meaningful but patterns irrelevant. Both, overlooking foundations and evidence, are untenable. © 2000 Published by Elsevier Science Ireland Ltd. All rights reserved.
Keywords:Neural coding; Point processes; Coding roles of spike trains; Foundations and evidence
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1. Introduction
This communication is intended as no more than a simple introduction to a Workshop, that,
as the one in Osaka in 1999, is on neural coding and spike trains. Its primary goal is to identify the topic’s foundations (Section 2). This is comple-mented by listing the salient overall highlights of the relevant data base (Section 3). Keeping both (Section 2) and (Section 3) in mind is indispens-able for strict consideration of neural coding and spike trains. This kind of purpose can be served acceptably, it is felt, by a succinct, almost tele-graphic text and by citations reduced to a bare minimum.
* Present address: Department of Neurobiology, University of California, Los Angeles, CA 90095-1763, USA. Tel.: + 1-310-8259582; fax: +1-310-8252224.
E-mail address: segundo@neurobio.medsch.ucla.edu (J.P. Segundo).
1Supported by Trent H. Wells jr. Inc.
2. Foundations
Foundations are the notions upon which the topic stands and is supported (e.g. Perkel and Bullock, 1968; Segundo, 1970). They jointly in-clude the topics core concepts (Section 2.1) and two critical experimental issues (Section 2.2).
2.1. Core concepts (and definitions)
A ‘coding’ or ‘encoding’ is a relation R be-tween two entities (variables messages etc.); the relation R is such that it can be summarized by rules, and these constitute its ‘code’ (Segundo, 1970). ‘Relatio’ has the elementary sense of a pair of objects in a definite order (Stoll, 1961); one-to-oneness is not required. ‘Coding’ there-fore has the sense used by Khinchine (1957) when referring to the conversion of source out-put alphabets to channel inout-put alphabets. ‘Asso-ciation’, ‘mapping’, ‘representation’, ‘transform-ation’, ‘’correspondence’ etc. are acceptable syn-onyms.
Codings can be summarized by codes, and thus are not random relations. A ‘random rela-tion’ is one that, in the rigorous sense defined by Chaitin (1975) allows no abbreviation (or, inpractice, only limited ones): this sense judi-ciously formalizes the everyday meaning of lack-ing plan or purpose. Codlack-ings imply bonds of some kind between the variables which, there-fore, covary significantly.
The fact of the coding includes the entities that participate, the variables that pertain to those entities, the states and/or changes matched experimentally, as well as the rules that, inferred from the data, tell us how the partner variables covary and thus compose the code.
The pertinent literature is quite clear in that codings are considered relations with bonds. The existence of a relation with a bond is, in fact, the indispensable take-off point, that which justifies interests in the topic and without which conceptually no issue relevant to it can exist and make sense. This must be, therefore, the natural and intuitive sense wherein neuroscientists per-ceive the notion.
2.1.1. Special codings
The coding R is a general abstraction that embraces all conceivable ways in which each partner can be described, and all conceivable sit-uations in which partners can be matched. Sub-sets of R constitute special codings. Each subset is defined with a particular restrictive criterion; different special codings can overlap. Codings are ‘neural’ when one partner at least involves the nervous system; only codings involving spike trains are dealt with here. The other partner is called the ‘referent’ (see Section 3.2).
(i) The natural coding RL is operant during the animal’s natural life; (ii) The experimental coding RE is that inferred experimentally: it in-cludes only particular features and particular sit-uations (e.g. interval histograms, respiration and phrenic neurons); (iii) Conditional codings (RN/ m* or RM/n*) arise when a particular condition-ing event (or class of events) in the space of one partner is chosen and its associations in the other space are considered. R may involve stim-uli to the skin and trains in certain neurons, respectively, events m in space M and n in N. One conditional coding associates prospectively a particular stimulus (event m*) and all trains (n) it could elicit; another one associates retro-spectively a particular train (event n*) and all stimuli (m) that could elicit it. Similarly, condi-tional codings arise in motor and other func-tions.
2.1.2. Partial aspects
All relations, and therefore all codings, exhibit partial aspects which inherent to them, must al-ways be present. Such aspects can be highly sig-nificant functionally. Understandably, papers generally tackle preferentially one aspect and pay at best superficial attention to others.
Using a single partial aspect to define coding is restrictive and thus not appropriate. Thus, the present author differs with his friends and col-leagues Perkel and Bullock (1968) and Perkel (1970) who defined coding as ‘the representation and transformation of information’. As pointed out, many papers tackle preferentially aspects other than those dealing with information.
mechanisms. These are the basic processes that allow the coding (e.g. those in receptors, neurons, synapses or effectors); (ii) Operational roles. These imply how the coding participates in each special network and function (e.g. in some spinal circuit or in vision); (iii) Reciprocal knowledge. The expression refers to the fact that, because of the coding, each partner has something to say about the other (e.g. sensory stimuli and the associated trains). Reciprocal knowledge can be quantified in several ways, for example using cross-statistics (e.g. correlations, spectra); it is quantified more precisely and exhaustively, using the specific measure called ‘mutual information’. It is necessary to keep in mind that valid investi-gator-centered issues raised inR may differ from meaningful neuron-centered ones participating in RL.
2.2. Critical experimental issues
Two experimental issues contribute to the top-ics foundations. One is the cardinal criterion used for evaluating spike trains (Section 2.2.1); the other is a finding that became seminal (Section 2.2.2)
2.2.1. The cardinal criterion for e6aluating spike
trains
Spontaneously and instinctively, neuroscientists evaluate individual trains by recognizing spikes individually and then noting when they occur: i.e. noting how whether spikes are many and packed or few and dispersed evolves as time advances and more spikes appear. Thus, the indices of the cell’s activity are the numbers of spikes it generates and the intervals between them, and both are pursued together as they unfold. In other words, neurosci-entists spontaneously and instinctively evaluate trains as series of events along time assimilating them to realizations of point processes (Cox and Lewis, 1966; Cox and Isham, 1980): the point process is the set of instantstI when spikes occur called ‘timing’. The nomenclature used here was proposed by Segundo et al. (1968).
In this context, therefore, a train, its point process description and its timing become syn-onyms and are indistinguishable. Accordingly,
any statement about a spike train so evaluated must imply both constitutional variables of all point processes: namely, numbers (or counts) translating into rates and intervals.
Because of this, it also is unavoidable that any putative code involving trains so evaluated must comply with two conditions: (a) In the first place, that code must jointly be a ‘number (count) or rate code’ and an ‘interval code’. Theoretically, for a particular train one can obtain the statistics and distributions of counts given those of inter-vals, and vice-versa. For example, the average rate and the average interval are reciprocals and inter-changeable, providing, of course, that they are estimated under identical conditions. This does not mean that the restricted descriptions obtained in practice necessarily are always equivalent; (b) in the second place, any such code must jointly include covariation of referent steady states with train overall averages, and covariation of referent fluctuations with train dispersions and sequences. A train’s dispersion and sequence are called its ‘pattern’: patterns are quantified by innumerable rate and interval statistics.
Logically, accepting the functional significance of average rates and their differences under some conditions, necessarily lead to accepting the func-tional significance of patterns and their differences under the same or other conditions. Indeed, as-sume that in some coding over a span (0, S] the average ratesr1andr2differ functionally and that patterns are irrelevant; necessarily, then, over (0, 2S] at the same average (r1+r2)/2, the patterns ‘r1/(0, S], r2/(S, 2S]’ versus ‘r2/(0, S], r1/(S, 2S]’ must differ functionally.
2.2.2. The seminal finding
Experimentally, electrophysiologists proved early that spike trains viewed as embodiments of point processes often covaried with functional referents, in other words often participated inneu-ral codings. This, acknowledged immediately, opened the field. Nowadays, it is accepted unani-mously, and almost unthinkingly taken for granted.
have been otherwise (Section 2.2.1), is accepted less readily (Section 3.5).
The above statements and conclusions are based on either logic or inferential rules: they are, there-fore, comprehensive and of a general nature. This does not imply they must hold also in every particular observation, regardless of where, when and how monitored and evaluated, i.e. rates and patterns will participate somehow in every coding, but each coding may well include local observa-tions where one or both do not participate.
3. Highlights of the data base
The following will first underscore what are considered the highlights of, i.e. the main conclu-sions inferred from, the relevant experiments (Sec-tion 3.1, Sec(Sec-tion 3.2, Sec(Sec-tion 3.3). This is followed by statements that involve facts as perceived, as well as value judgements about them (Section 3.4, Section 3.5). All are stated telegraphically.
3.1. Confirmation o6er a 6ast domain of the
seminal finding (Section2.2.2) that spike trains as point processes and timings participate in neural codings
Particular instances of functionally meaningful codings matched spike trains with referents in practically all fields explored (see Section 3.2). As could not be otherwise (Section 2.2.1), all codings included numbers (counts rates) intervals, averages and patterns.
3.2. Demonstration of general and of synaptic codings, each with characteristic kinds of referents
3.2.1. General codings
Referents are sensory or effector parameters, normal or abnormal states, etc. (e.g. tactile stimuli, respiratory patterns, sleep states, seizures).
3.2.2. Synaptic codings
Referents are trains in neurons having direct synaptic connections with the first. Connections are at either the input or output side of the targeted
neuron: the latter acts simultaneously as, respec-tively, post or presynaptic partner. They provide the operational unit for nervous systems conceived as networks of neurons that generate spikes and interact via synapses.
Synaptic codings imply issues and notions whose recognition and dissection are critical for under-standing network dynamics (e.g. Bullock, 1961; Segundo and Perkel, 1969; Segundo, 1970). For example, in synaptic codings, the postsynaptic cell adjusts its output trains to the input presynaptic spike trains: it can thus be said to perform as an ‘analyser’ device for the former. When dissimilar arriving trains associate with dissimilar output trains, the fact of a difference is preserved: the postsynaptic neuron thus ‘reads’ that difference discriminating between trains. The essence of inte-gration is that pre and postsynaptic trains, though related, are different (excepted are special cases such as neuromuscular synapses).
Moreover, the postsynaptic neuron unceasingly resolves the alternative of firing or not firing, thus acting as a ‘decision-making element’. Its decisions are based on happenings within a bounded recent epoch. Within this period, a veritable memory, the postsynaptic neuron evaluates the rate and interval averages and patterns of recent pre and postsynap-tic spikes. Epochs and spikes therein have been alluded to as ‘integration period’ and ‘influential events’, respectively.
3.3. Disclosure of a remarkable o6erlap between
the train features that participate in general codings and the features seen in synaptic codings
Because of this agreement, networks can use trains as tools in their ongoing dynamics; its importance is, therefore, paramount.
3.4. Introduction of neural coding and spike trains into its natural formal contexts
comprehen-sion. The present Workshop will add significantly to it.
In particular, the general concepts of Commu-nication and Information Theories apply to all systems formed by interacting entities (e.g. Se-gundo, 1970; Rieke et al., 1997). This includes neural codings individually, as well as entire ner-vous systems composed by endless numbers of such codings. Application of those concepts to particular cases has not always been as fruitful as one could wish: this is because of two main reasons. One, that our often limited knowledge of neural function hinders pursuing what Shannon called semantic and effectiveness issues; these is-sues are self-evident in sensory and motor neurons where main roles are, respectively, identifying stimuli and eliciting particular contractions. An-other reason is that precise quantifications using entropies and mutual informations demand know-ing all outcomes and havknow-ing reliable estimates of all probabilities: this often requires very large data sets.
3.5. Untenable hypothesis
On the negative side, a measure of confusion arises from a tendency to overlook the topic’s foundations (Section 2.1, Section 2.2) – notions, experimental issues – plus the evidence made avail-able subsequently (Section 3.1, Section 3.2, Sec-tion 3.3), as well as their logical frameworks and the guidelines from them derived. This neglect characterizes two opinions, currently expressed or implied. Namely: (i) the trenchant opposition be-tween rate and interval statistics; and (ii) the exclusive significance of overall average rates plus the irrelevance of intervals and of local rate pat-terns. Both opinions clash with foundations and evidence, and thus are untenable; opinion (ii) also
is inconsistent intrinsically (see Section 2.2.1). Publications espousing such credos preserve a fac¸ade of verisimilitude by practicing biased cita-tion policies whereby references espousing them are accepted and all others purged. Their not negligible support probably reflects issues emerg-ing from the Sociology of Science.
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