Brain functions and neural dynamics

J. Theoret. BioL (1970) 26, 93-120

Brain Functions and Neural Dynamics t E. M. HARTH, T. J. CSERMELY:~, B. BEEK AND R. D. LINDSAY

Department of Physics, Syracuse University, Syracuse, N. Y., U.S.A. (Received 20 February 1969) Anatomical and physiological evidence is cited for the existence in the CNS of more or less discrete populations of interconnected neurons. These are given the term netlets. A model based on these observations is presented, in which it is assumed that the netlets are the fundamental building blocks out of which nets of considerable complexity may be assembled. The connectivity within each netlet is assumed to be random. Neuronal macrostates are defined in which the fractions of neurons active in each netlet are the dynamical variables. Thus the temporal and spatial fine structure of neuronal activity are considered to be of secondary significance and are disregarded. These assumptions bring about an enormous reduction in complexity. Thus calculations and computer simulation studies become possible for systems hitherto inaccessible to quantitative description. It is hoped that the features retained in the model play a sufficiently significant role in the functioning of real neural nets to make these results meaningful. The mathematical formalism and detailed numerical results appear in another paper of this issue (Anninos, 1970). Some of these results are anticipated in this paper and their implications for our model are discussed. The study proceeds from' a treatment of isolated probabilistic netlets to the dynamics of interacting netlets. Of particular interest are the conditions under which a netlet will go into sustained activity and the often extremely delicate control exerted by afferent excitatory or inhibitory biases. Hysteresis effects are common and may represent a type of shortterm memory. A variety of neural functions are listed to which some of these mechanisms may be applied. A m o n g these are the modulating effects of the brain stem reticular formation on cortical and spinal neuron populations ? The research reported in this paper was sponsored by the Aerospace Medical Research Laboratories, Aerospace Medical Division, Air Force Systems Command, WrightPatterson Air Force Base, Ohio, under contract No. F 33615-67-C-1413 and the Office of Naval Research, contract No. ONR N00014--67-A-0378-0001, with Syracuse University Research Institute. Further reproduction is authorized to satisfy needs of the U.S. Government. ~t Present address: Physiology Department, State University of New York, Upstate Medical Center, College of Medicine, Syracuse, N.Y., U.S.A. 93

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and the "energizing" of cortical centers by spontaneous activity in sensory systems. Finally the concepts of netlet interaction are applied in conjunction with the principle of synaptic facilitation to information processing in the cortex. Examples given are sensory-sensory cortical conditioning and the formation of the classical conditioned reflex. 1. Introduction

In spite of the many recent advances in neuroanatomy and neurophysiology, no one today is in a position to say precisely what the brain does, and how it accomplishes its tasks. It is perhaps this failure of the biological sciences --the tremendous efforts notwithstanding--that has invited a host of not always welcome outsiders to try the methods of their particular specialty on the problem. Possibly, the intrusion is justified by the nature of the system. Unlike other organs, the brain exhibits none of the simple physical principles on which even a rudimentary hypothesis of its functioning may be based: it is neither a pump, nor a filter, nor a chemical factory, resembling rather some of the most complex dynamical systems considered by physicists, or logical structures studied by mathematicians. It is generally believed that the key to understanding brain function is to be found in one component making up the brain: the neurons. Consequently, most theoretical studies begin with a consideration of systems of units bearing a functional resemblance to neurons. These remarkable cells are uniquely adapted to the processing of information and the performance of logical tasks. Their analogy to logical elements in a computer was stressed by von Neumann (1958) and the logical capabilities of a network of neuronlike elements were explored by McCulloch & Pitts (1943) and Kleene (1956). We shall follow the approach of considering the individual neuron as a decision-making element which goes abruptly from a resting state into a firing state in an all-or-none manner, The transition occurs whenever the combined inputs from other neurons exceed a threshold value. Detailed properties are described by parameters such as threshold, absolute and relative refractory periods, summation time and synaptic delays, which shall have the usual meanings. The manifestation of neuronal activity is the action potential, which is propagated without attenuation along the axonal fibers whose terminals, the synapses, mediate (by chemical action) the inputs to other neurons. These inputs are changes in the membrane potential of the post-synaptic cell, taking it either closer to or farther away from the firing threshold. Physiologists speak of excitatory and inhibitory post-synaptic potentials. Connections among neurons are manyfold. A single cortical neuron may have many tens of thousands of incident synaptic junctions. The brain thus

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has the aspect of a communications network of enormous complexity. At this point the theorist who attempts to gain insight by the construction of a model faces the problem of deciding which structural and dynamical features of the system to stress and which to ignore. If his choices are fortunate, he will be able to explain some properties of the biological counterpart of his model. It must be stressed that the model can never be equivalent to the real thing, nor can it be used to explain all features of interest. A large number of different models may be necessary for a complete description; they must be based on different sets of assumptions. In purely physical terms, the brain could be described by detailing the neuronal circuitry and describing the resulting dynamics. Such a description would still be incomplete, however, for one should also be interested in the functional correlates of neuronal activity, that is to say, the observable sensory and motor events, as well as the generally unobservable sensations which correspond to given physical states of the system. Thus, in addition to the structural and dynamical assumptions of the model, we must sooner or later face what we may call the semantic problem of neural activity. The human brain is estimated to contain approximately 3 × 101° neurons, most of these making up the cerebral cortex. Only fragmentary knowledge exists concerning their connectivity, and many fundamental questions, such as the nature of the physical changes that constitute learning, remain unanswered. The dynamics of large numbers of interconnected neurons has been studied only for some highly simplified nets, generally disregarding any of the known structuring. A description of the dynamics of such a net may consist of the specification of the firing times of all neurons. The description of the neuronal state of the human brain at any given moment would thus require 3 × 101° bits of information and the specification of the dynamics of the system would involve 3 × I0 t° time dependent variables. Such an approach would preclude consideration of more than the minutest portion of the neural net. Clearly if a holistic description of the system is to be achieved, one must reduce drastically the number of parameters to be used. A necessary condition for advance lies therefore in uncovering the appropriate macrostates into which the large number of microstates may be grouped. The key word here is "appropriate", and requires some explanation. The macrostates of a gas are described by the readily observable thermodynamic variables pressure, volume, and temperature. In the dynamics of neural nets the activity of any single neuron is in principle observable, but the simultaneous monitoring of more than a few neurons presents great technical difficulties. If the detailed firing records of a large number of neurons during a time interval At, or even the simultaneous firing states of these neurons at

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t, are taken to be the microstate of the system, then these microstates are as unobservable as the microstates of a thermodynamic system. On the other hand, the EEG which is affected by the activities of large numbers of neurons could in principle be taken to define a set of macrostates and corresponding thermodynamic variables. However, the requirement of appropriateness of macrostates goes beyond the question of observability; to be useful, they should be what has been termed the neuronal correlates of specific brain functions. These functions, in spite of their complexity, can be labeled and categorized. Sensory events such as the perception of a buzzing sound or the recognition of a familiar face, motor events such as walking, scratching, etc., are macrostates, each having a very large number of possible complexions. The neuronal macrostates we seek, then, are the correlates of sensory and motor events (and higher brain functions, to the extent to which they can be described). It should by no means be taken for granted that such neuronal macrostates must exist, nor that their description can be made sufficiently simple to be of practical value. However both will be assumed here. It is difficult to see how significant progress in understanding the brain can ever be achieved unless these two assumptions are justified. It would seem that EEG patterns are good candidates for our macrostates, being both observable and showing at least some correlation to brain function (Morrell, 1967). We wish to add the further requirement, that a simple functional relationship exist between the macrostates and the (unobservable) microstates. This is clearly the case in the kinetic theory of an ideal gas where the thermodynamic variables are derivable from the coordinates and momenta of the molecules. The relationship between the EEG record and the firing pattern of individual neurons unfortunately is not precisely known. For this reason, EEG patterns do not lend themselves readily to a description of the dynamics of neural activity. Since we wish to make neural dynamics an integral part of our study, we must look for a different description of the system. We can now give in very general terms the long-range aims of our efforts: (1) to choose models of neuronal connectivity which preserve certain design features observed in the brain; (2) to specify for such systems, sets of variables describing appropriately chosen macrostates; (3) to study the dynamics of these systems in terms of the variables chosen above; and (4) to relate the neural mechanisms observed in this way to specific brain functions. The present paper is concerned with items 1, 2 and to a lesser extent, 4 in

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this list. Item 3 is discussed in another paper in this issue (Anninos, Beek, Csermely, Harth & Pertile, 1970) which will be referred to as II. In a previous paper (Harth & Edgar, 1967), a semantic hypothesis of neural activity was discussed. The relevance to the present work will be discussed further. 2. Biological Basis of Model In this part we wish to discuss some direct and indirect evidence supporting a model which we hope incorporates functionally significant features of neuronal organization, while retaining enough simplicity to make possible detailed considerations of its dynamics. The point-of-view to be developed places emphasis on the observed differentiation of the neural mass into pools of functionally more or less equivalent neurons. The concept of neuron pools is not new. We know of such sets of neurons forming functional units in the spinal cord, the so-called motoneuron pools. Hebb (1949) has suggested that functionally equivalent neuron groups, which he called cell assemblies, exist in the cerebral cortex and are responsible for learning. It has long been known that specific sensory or motor events may be simulated by stimulation of particular cortical areas by means of a small electrode (Penfield & Rasmussen, 1955). The effect is most pronounced in areas surrounding an epileptogenic focus, i.e. a region rendered hypersensitive by some pathology. The electrodes used do not allow the experimenter to select single neurons, but distribute the electrical stimulation over areas of several square millimeters, comprising probably of the order of thousands of neurons. Yet the elicited sensory or motor event is quite definite and generally reproducible. It may consist of the flexion of a particular muscle (for stimulation in the motor cortex) or the perception of such visual elements as stars, flashing lights or moving colored spots (for stimulation in the visual cortex) or sensations described by the subjects as clicking, buzzing, or rumbling noises, when the auditory cortex was stimulated (Penfield & Roberts, 1959). Other experiments have demonstr~ed that when single cortical neurons of the appropriate centers are monitored, there appears to be no invariant relationship between such unit activity and repeated sensory input (Jasper, Ricci & Doane, 1960) or between unit activity and a very definite, repeated motor function (Doty & Bosma, 1956). Morrell (1961) expressed this apparently random unit response this way: "From the cellular point of view it would appear that a particular performance utilizes multiply represented pathways and that the same behavior is not achieved each tim by means of the same cells or the same pattern of interaction".

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There now exists ample evidence that the establishment of memory requires a phase of sustained neural activity in stimulus-specific neuron populations. Aspects of such reverberatory activity are discussed by John (1967). If neural activity is seriously disturbed, e.g. by electroshock, shortly after a sensory event, this event is lost to permanent memory (Quartermain, Paolino & Miller, 1965). On the other hand, the administration of so-called analeptic drugs like strychnine (Pearlman et al., 1961) and picrotoxin (Breen & McGaugh, 1961), which tend to increase neural activity by blocking inhibition, was found to facilitate memory storage. These experiments suggest that the rate of memory consolidation depends on the level of neural activity. Direct evidence for a high degree of redundancy in unit response was discovered by Mountcastle (1957), who observed that neurons that are located along radial columns in the somato-sensory cortex exhibit identical receptive fields. The fact that the same latencies and nearly identical initial patterns of discharge appear anywhere within such columns, led Mountcastle to consider them as "elementary units of organization". Similarly it was shown by Asanuma & Sakata (1967) that discrete colonies of neurons in the motor-sensory cortex of the cat can be identified, such that neurons belonging to a colony have projections terminating in the same motoneuron pool. In a subsequent study (Asanuma, Stoney & Abzug, 1968) it was shown that the same cortical neuron colonies were also characterized by specific receptive fields. Thus the neurons within a given colony exhibit a high degree of similarity both with respect to their efferent and afferent specificity. In a long series of experiments, Hubel & Wiesel have investigated the visual system of cats as welI as primates. It was observed that the neurons in the visual cortex are not the simple spot detectors found in the retina (Kuffler, 1953) and later in the lateral geniculate nucleus (Hubel & Wiesel, 1961) but represent more complex cognitive elements such as directed lines, edges, corners, and even moving edges (Hubel & Wiesel, 1962). It was established also that these functions, and the corresponding neural organization, were present at birth rather than being the result of learning (Hubel & Wiesel, 1963). One of the remarkable features found was the fact that in general the orientation of the linear receptive fields were identical for all cells contained within a columnar structure perpendicular to the pial surface. Commenting on the findings of Hubel & Wiesel, Colonnier (1966) pointed out that the afferent sensory fibers may go only to some of the cells in the column, specifically the stellate ceils of layers III and IV. The uniform response of all neurons within a column, both in the sensory-motor and the visual cortex, would then be the result of local spreading of neuronal activity.

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Such radial spreading was observed directly in intracortical microstimulation (Stoney, Thompson & Asanuma, 1968); it was shown also that the effect of such stimulation fell off very rapidly with distance in a plane parallel to the pial surface. Anatomical evidence is provided for the radial spreading of activity by the "climbing" fibers of fusiform stellate cells and the radial axons presumably activating other stellates (Colonnier, 1966). Neuroanatomical data complement the above physiological findings. The strong vertical organization and the relatively short-range lateral connections in the cortex were described by Sholl (1956) and more recently by Colonnier (1966, 1967). There appears to be some evidence also for a periodicity in cell and fiber density suggesting that these may be the anatomical analogues to the functional columnar organization discovered by Mountcastle & Hubel and Wiesel. Little is known to date either about the anatomy or the physiology of large portions of the cerebral cortex often referred to as association areas or uncommitted cortex. Golgi-stained sections of these areas show that local neural organization is at least not strikingly different from what is observed in the sensory and motor areas, in that connectivity again favors the radial direction. Electrode stimulation of exposed areas in the association cortex which lie near an epileptogenic focus (Penfield & Perot, i963) can produce distinct and reproducible sensations, as was observed at lower cortical levels, but this time they are much more complex, consisting frequently of detailed flashbacks of previous experiences. The striking fact we wish to underline in all these examples is that simulation or recall of more or less complex sensory events does not require an incident frequency code (stimulation is accomplished generally by a low voltage 60 cycle wave) nor a delicate spatial patterning of incident excitation. It is sufficient, apparently, to set up neuronal activity in a cortical region specified only to within about a millimeter. Cerebral deficiencies resulting from lesions in the association areas may provide some clues concerning the organization of this part of the cortex (Luria, 1966). In general these defects (aphasias, agnosias) concern functions at a higher level of integration, for example the recognition of faces, without impairment of the more primitive sensory faculties. They have been referred to as associative sensory blindness, a modernized version of the German original "Seelenblindheit" (Lissauer, 1889). Such evidence has led Konorski (1967) to postulate the existence of gnostic units in the association cortex, these units being presumably analogous to the more elementary sensory units found by Mountcastle and Hubel & Wiesel. We shall assume, in developing our model of cortical functioning, that these gnostic units also consist of groups of neurons, integrated into functional units by strong local interconnections. T.S.

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3. Design and Randomness Before using these data in the selection of a particular description of neural activity, we wish to ask this question concerning neuronal structure: is the connectivity of the neurons deterministic or random ? The controversy between these two viewpoints concerning cerebral organization has a long history, and may be compared with the classic controversy over the wave or particle nature of light in physics, The latter was finally resolved not by the victory of one and the defeat of the other, but by a reconciliation of the seemingly irreconcilable points of view--the principle of complementarity. Unquestionably the brain has structure, which is present at birth, serves important biological functions and is presumably genetically determined. In recent years evidence has been accumulating for the existence of highly specific, inborn neural circuits in many parts of the cerebral cortex, in seeming contradiction to the mass action principle and the concept of cortical equipotentiality (Lashley, 1931, 1933, 1950). The similarity of the design principles extant in nearly all sensory systems (v. Brkrsy, 1967) makes it tempting to postulate that this kind of genetically determined neural organization is characteristic of the entire neocortex. An extreme view of this sort would be the statement that every synaptic connection throughout the nervous system is genetically determined. It has been argued that the limited information content of the DNA molecule precludes such a view and that therefore elements of randomness must be present. This is not true; it is clearly possible to specify the connectivity in every detail and yet use only a very limited amount of information, in the same way in which we can describe the space coordinates of large numbers of atoms in a crystal merely by giving the parameters of the lattice. Thus, any symmetry principle that can be found reduces the amount of information required for a detailed specification of the system. Symmetry principles are information-saving devices. In the case of neural connectivity in the higher cortical areas we know as yet too little about symmetry laws to be able to tell whether all of the remaining information content of a particular connectivity can be supplied by our genes. The other extreme view, that all neural connections in the cortex are probabilistic, is clearly contradicted by facts. However, the possibility that details of the cortical connectivity contain elements of randomness can certainly not be ruled out. The high degree of neural specificity that is frequently observed in the peripheral nervous system (Edds, 1967) probably does not exist in the cortex. Accordingly it will be assumed in this work that the neural net in its central portions is characterized by genetically determined design in the large, while the local connectivity is largely probabilistic, subject only to certain statistical design parameters.

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It alSO appears reasonable to assume that the connectivity of the cortex in the primary sensory and motor areas has a higher degree of design and organization than the large non-specific areas called the association cortex. It is observed that in almost every case biological design is coupled with a definite biological function, and that, where the design is genetically determined, the function is of genetically predictable value. But the individual, man in particular, is faced with many tasks for which evolution could not possibly have equipped him, such as the necessity of stopping a car at a red light. What neural circuitry is to take care of these genetically unpredictable tasks ? It has been argued that there exists in the brain an immense pool of specialized circuitry capable of taking care of every possible exigency, and that circuits are activated as experience requires. It was shown by Bremermann (1967) that the "genetic cost" of such a system would be staggering. On the other hand, it was shown by Harth & Edgar (1967) how a completely random net can become structured as a result of sensory inputs and perform a variety of learned tasks. We propose therefore that design and randomness in cortical architecture, in so far as they are separable, correspond to complementary functions, namely the performance of genetically predictable and unpredictable tasks respectively. Accordingly, we assume that the portion of the cortex whose task is to carry out fundamentally unpredictable tasks--the so-called association cortex--is a relatively unstructured, probabilistic net. This division into two components of complementary properties may correspond to a more gradual transition between design and randomness in the real brain. Most likely, no areas there are completely designed nor completely probabilistic. 4. The Netlet Model We now present a model of greatly reduced structural complexity and select a set of variables to describe what we called above an appropriate set of macrostates of the system. (A) SPECIFIC ASSUMPTIONS OF THE MODEL (1) Neuronal organization in the sensory and motor portions of the cortex is described as a collection of many small discrete nets of neurons; these will be called netlets. (2) Each netlet forms a dynamical unit. Neuronal activity within a netlet may be endogenous or triggered by afferent fibers. Such activity may be transient or self-sustained.

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(3) Neural connectivity is assumed to be probabilistic in the small, but subject to design in the large. This means that connections within a netlet are chosen at random. Likewise the distribution of afferent fibers within a netlet is assumed to be random. A set of parameters which will be defined in II describes both the internal and external connections for each netlet. (4) We define a level of activity within a netlet as the fractional number of neurons firing at a particular instant. It is assumed that this is the only dynamical variable of interest. Details of the spatial distribution of activity within a netlet will be disregarded. (5) Netlets can be labeled according to the particular motor function they control, or the particular sensory event which constitutes the chief afferent. The neflet concept is best illustrated by a graphic description of net structure used by Harth & Edgar (1967). This description is a schematic of the so-called connectivity matrix {ki~} which is defined as follows: Let kij be the change of membrane potential at the axon hillock of neuron i resulting from the firing of thejth neuron. This quantity will subsequently be referred to as the post-synaptic potential (PSP). Thus k~j is the net effect of all the synapses for which the ith neuron is the post-synaptic and the jth neuron the pre-synaptic cell. We call this the coupling coefficient from the jth to the ith neuron. A coupling coefficient of zero is to denote the absence of a synaptic link. Note that all connections are one-way connections. The coupling matrix {k~j} is therefore in general an asymmetric matrix. Although our computer simulation programs allow us to consider the values of the elements k~j to be distributed statistically, it is assumed in this paper that all non-vanishing elements within a netlet have the same value. In Fig. l(a) we show a schematic diagram of a hypothetical neural net. There each dot represents a non-vanishing element in the matrix {k~j}, or a connection from the j t h to the ith neuron. The net shown is composed of three netlets, labeled A, B and C. The three diagonal blocks thus represent the internal connectivity of the netlets; the randomness of the points reflects what we called the randomness in the small. The off-diagonal blocks represent the coupling between netlets; the details of this design in the large are again probabilistic. An equivalent diagram showing the three netlets and their external couplings is shown in Fig. l(b). The dynamics of homogeneous, randomly connected neural nets has been studied by many authors. Some of this work will be reviewed briefly and extended in II. It is felt that with some simplifying assumptions concerning the characteristics of individual neurons, the behavior of such nets is by

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now fairly well understood. Important extensions to these studies, incorporating more complex features such as relative refractory periods, spontaneous activity, temporal summation, adaptation and others, can be treated by computer simulation and are being pursued at this laboratory. In the model outlined above, homogeneous probabilistic nets are used as building blocks to simulate known design in the large and presumed randomness in the small. Such systems of interconnected netlets will be called compound nets. The dynamics of such systems is difficult but tractable. We have begun the study of some very simple systems and uncovered some neural mechanisms which may well play important roles in brain functioning. /1

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We believe the netlet model to be a useful description of at least certain portions of the cerebral cortex in view of some of the neurophysiological and anatomical facts cited in the preceding section. The differentiation into cell colonies is certainly one .of the more striking features of neural organization. We refer again to the many observations of Penfield and co-workers which suggest that temporal and microscopic spatial details of activity play a secondary role, thus lending support to our assumption (4). This is reinforced by the experiments (Jasper et al., 1960; Doty & Bosma, 1956) showing the non-repeatability of neuronal microstates. The radial spreading of activity (Stoney et al., 1968) and the observed vertical dendritic and axonal structure (Colonnier, 1966) are evidence that the neuron colonies found by Asanuma & Sakata (1967) in the motorsensory cortex or the columnar structures found by Hubel & Wiesel (1965) in the visual cortex are in fact nets of neurons, or what Mountcastle (1957) called "elementary units of organization". Finally, some, admittedly much weaker, evidence was cited, hinting that the netlet organization may extend beyond sensory projection areas and the motor cortex into the so-called association areas.

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It should be pointed out also that the netlet concept may be applied not only to spatially differentiated neuron populations. There exists evidence (Miller, 1965; Kety & Samson, 1967) that different central neurons may respond to different chemical transmitters and hence can be triggered only by neurons emitting that particular transmitter. Neurons may also be distinguished on the basis of their excitability. Thus Burns (1958) refers to a "type B network" of highly excitable cells in the cerebral cortex. Such cell-type populations, if interconnected, may also be treated by the netlet formalism. The enormous simplification of both the dynamics and the description of neural activity which results from the application of the model should be obvious.

(B) DYNAMICS OF NEURAL NETS

In this section we present in condensed form some of the results of the calculations and computer simulations which are described in II. The dynamics of randomly connected homogeneous neural nets form the necessary starting point. In our treatment, we follow the customary procedure of quantizing time in units of the synaptic delay z, and assuming that the neurons can fire only at times which are integral multiples of z. Most of the calculations were carried out under the assumption that all neurons have absolute refractory periods in excess of z but smaller than 2~. This assumption, because of its mathematical simplicity, is a convenient starting point. It is also not unreasonable considering that refractory periods are generally found to be close to 1 msec and synaptic delays in the neighborhood of 0.5 msec (Eccles, 1964). No relative refractory periods were considered; the firing thresholds were returned to normal at the earliest time a neuron could fire again, 2~ after a previous firing. Dendritic and axonal transmission delays over small distances are assumed to be negligible compared with synaptic delays. With these assumptions we can define a quantity ~, which describes the fractional number of neurons firing at time t = nv. Numerical results on the dynamics of nets are obtained by direct computation or, for the more complex phenomena, by computer simulation.

Single isolated netlets Structural parameters of netlets include the fraction of inhibitory neurons (h), the average number of neurons receiving post-synaptic potentials (PSP's) from an excitatory neuron ~+), the average number of neurons receiving PSP's from an inhibitory neuron (#-), the PSP produced by an excitatory neuron (x+), the PSP produced by an inhibitory neuron (x-),

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and the minimum number of EPSP's required to trigger a neuron if no IPSP's are present (r/). A complete glossary of terms is given in II. We have assumed for the purpose of all the calculations in this paper and I I that the absolute refractory period of a neuron lies between one and two synaptic delays and that the summation time is less than one synaptic delay. With these assumptions, problems o f conditional probabilities are avoided since the activity 0c,+~ in an isolated netlet is entirely determined by the preceding activity ~.. The dynamics of single probabilistic netlets without external connections and without spontaneous activity is discussed in II. We give below some of the important results. In Fig. 2 are shown a series of typical curves of c¢,+~ vs. ~. for various thresholds r/. We distinguish between three prevalent modes of behavior which we label class A, B and C, respectively. There exist other modes, some of which were discussed by Smith & Davidson (1962a,b). The curve cor-

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responding to r/ = 1 in Fig. 2 characterizes a class A net. It has the property that for low activities cq the subsequent activity ~n+l will always be larger than c~n, up to the point where the curve crosses the 45 ° line. This point, generally slightly below 0-5, is a point of stable activity. Such a net will clearly be unstable in its quiescent state since any small fluctuation will trigger it into sustained activity, which is expected to approach or oscillate about the point of stable activity. A class B net is defined as one in which there exists a threshold for being triggered into sustained activity. The curve r/ = 2 in Fig. 2 belongs in this category. It is seen that for very small values o f ~n (below point A in the diagram), an+l is smaller than an, hence activity will decay toward zero in this region. Point A is a point of unstable, B of stable equilibrium. The remaining curves in Fig. 2 are examples of class C nets. They lie wholly below the 45° line, hence activity will decrease for any initial value a n. Computer simulations gave excellent agreement with the computed data.

Time dependence of activity The mathematical formalism presented in II can be used to predict the development of activity in isolated netlets. Typical trajectories for a class B netlet with different starting activities are shown in Fig. 3. It is noted that damped oscillations are predicted about the value of the stable equilibrium. One would like to know something more about this kind of sustained activity. Although we have stated that the details of the activity within a netlet will in general be disregarded, it is of interest to know whether under IO

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conditions of sustained activity the sequence of the corresponding microstates is periodic, and if so, what is the length of the periods. The computed time dependence of activity was also checked by computer simulation for nets having no inhibitories. The agreement was excellent except for one feature. The simulated nets, instead of showing activity converging to the stable equilibrium value, exhibited activity which rapidly approached a condition in which activity oscillated back and forth between two subsets of neurons in the netlet. This highly non-ergodic behavior was unexpected at first since a very naive calculation would predict very long cycling periods. It is shown in II however, that, at least for nets having no inhibitory neurons, the observed behavior results from the fact that in the simulated nets one is dealing with random but frozen connectivities, while the simple statistical model on which most of the calculations were based neglects the coherence between successive states and, in effect, re-randomizes the connectivities at every successive interval. We believe that inhibitories and different assumptions concerning refractory and summation periods will make the activity more nearly ergodic. This question remains to be investigated.

Netlets with afferentfibers As the first step toward a dynamics of compound nets, we have investigated the behavior of netlets which are subjected to sustained, more or less constant, inputs from afferent fibers. These inputs may be excitatory, inhibitory, or a combination of the two. For the details of the assumptions and calculations the reader is referred to II. The striking aspect of this analysis is the appearance of hysteresis loops: a slow change of the level of afferent inputs leads to irreversible changes in the steady state activity of the netlet. This is best illustrated in a schematic diagram we shall refer to as a phase diagram [Fig. 4(a)]. Here a class C netlet is subjected to sustained excitatory input described by the parameter a. The quantity ~ss is the steady state value of the netlet activity obtained by requiring that c~.+1 = c~.. The arrows in the diagram indicate the direction of activity change following a fluctuation of activity away from the condition en+ 1 = e.. The curve can thus be divided into stable portions (solid line) linked by an unstable portion (dotted line). Figure 4(b) shows what happens if we change the level of afferent excitatory input quasi-statically. The resulting hysteresis loop has a lower and an upper reversible portion (solid lines) linked by the irreversible upward transition at o- - crcrit and the downward transition at a = ao. The steady input may be viewed as a bias imposed on the response of the netlet to single momentary inputs. Thus it furnishes a mechanism for ex-

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plaining the tonic effect of one portion of the neural system upon another. This is best seen if we consider a netlet with steady input between ao and tr=rtt, whose activity is initially at the lower of the two steady-state conditions. Referring to Fig. 4(a), it is seen that a momentary stimulation w ill cause a transition to the higher steady state if it causes the activity to rise above the dotted line at any time. The dotted line therefore represents the thresholds for triggering the netlet into the state of high sustained activity. It is seen that this threshold depends very sensitively on the level of the steady input o-. The high activity, once established, can be extinguished only by lowering the level of steady input below a o. In Fig. 5(a) we have combined in one phase diagram the effects of purely excitatory (a +) or purely inhibitory (a-) sustained inputs. It is seen that some class C netlets, labeled (;'2, have no unstable portions in their steady~

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state curves, hence no hysteresis. Class A and class B netlets are interesting, in that inhibitory inputs are necessary to extinguish the high steady-state activity once it is established. In a class A netlet the lower level steady-state activity is always zero but can be maintained only if the inhibitory input exceeds O'crit. ~t Results of varying the steady inputs quasi-statically are shown in Fig. 5(b). Netlets A, B, and C1 are seen to have hysteresis loops, while C2 behaves reversibly over the entire range. The diagrams in Figs 4 and 5 are schematic. Quantitative data on selected netlets are given in II.

Hysteresis and memory The behavior of class A, B and C 1 netlets belongs to the category of time-independent irreversible systems, discussed by Katchalsky & Oplatka (1966), which must be invoked as the physical basis for memory. Such systems are characterized by a hysteresis loop and the existence of more than one steady state for a given value of the external parameter. In our case this parameter is of course the level of steady input. Our system differs only in that transient states--due to momentary stimulation--may appear anywhere in the diagram, causing irreversible transitions if they exceed the threshold given by the metastable states. It was pointed out (Enderby, 1955) that the theoretical hysteresis loops of the type shown in Fig. 4(b) should give rise to a set of scanning curves which reflect the existence of domains similar to those observed as the Barkhausen effect in ferromagnetism. Simulation studies are now being performed to see whether simulated netlets show such domain effects. Spontaneous activity The apparently random firing of sensory neurons in the absence of sensory stimulation has been known since the early work of Adrian & Zotterman (1926). Although the origin of this phenomenon is far from clear it is now an established fact for most sensory systems. A possible significance of this random noise arriving at the CNS was pointed out by Granit (1962), who believes that such activity provided the necessary activation of cerebral systems. We believe that our model, particularly the calculations on netlets with sustained random inputs, provides a neural mechanism for such tonic effects. There exists a controversy over whether or not cortical neurons deprived of all afferents also exhibit spontaneous activity (Burns, 1958). But whether the activity is truly endogenous or the result of peripheral noise, the existence of spontaneous activity changes the dynamics of cortical activity from a deterministic to a stochastic process. It might, for instance, have the effect

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of preventing the short cycle repetitive activity we have mentioned and cause a more ergodic activity pattern. The dynamics of netlets subjected to internal spontaneous activity is treated in II. It is found there that such activity has an effect not very different from that of a sustained input into the net. Thus phase diagrams and hysteresis loops similar to those in Fig. 5 are obtained when tr is replaced by a random spontaneous firing rate. This process will generate activities in the netlet, which over a given range may have two distinct steady state values. Transitions may be induced from the lower to the higher state by momentary excitatory inputs, opposite transitions by inhibitory inputs. The system is therefore a resettable memory unit--the transitions are irreversible in the thermodynamic sense since they involve non-equilibrium states. The random spontaneous firing rate thus serves the important function of fixing the thresholds for these transitions.

Coupled netlets Calculations and computer simulations were carried out in II for pairs of nettets coupled as indicated in Fig. 6. In Fig. 6(a), netlet B receives afferents from A but does not in turn affect the activity in A. Thus A is an isolated netlet and we may determine the time dependence of its activity as outlined above. If this activity is constant or slowly varying, the formalism of the

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preceding sections may be used to determine the activity in B. In general these conditions are not satisfied and we must resort to computer simulations or compute the activities step by step. In Fig. 6(b) is shown a two-netlet loop. These structures often form reverberating systems with very sharp activation thresholds. Not surprisingly, sustained activity may occur even for coupled class C netlets. Such a situation is shown in Fig. 7 where two identical class C netlets are coupled symmetrically with excitatory connections. We have taken as the starting point an activity ~ol in netlet A and ~o2 = 0 in B. The time dependence of activity in A and B is shown for two different values of c~ol, one just below and one above the threshold for producing sustained reverberations in the loop. Note that activities quickly become identical in the two netlets. The curve in Fig. 7(b) suggests that subthreshold activity near the unstable equilibrium may be maintained for some time before it decays.

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Compound nets We have already discussed the connectivity matrix {ku} and shown how a neural net c o m p o s e d o f interconnected netlets m a y be represented by a diagram o f the kind shown in Fig. 1. In keeping with our assumption 3 concerning the randomness o f local connections it is convenient to express the connectivity in terms o f a matrix o f macroscopic coupling coefficients Kzm where l and m are netlet labels and Kz,, is a function o f the design parameters o f the two netlets and their interconnections. The diagonal elements Kn o f this matrix are called the self-coupling coefficients, the off-diagonal elements are the mutual coupling coefficients (Csermely, I968). Given {K~,,} and initial conditions o f the net, one can in principle c o m p u t e the activities o f each netlet at any subsequent time interval. While such computations m a y still be quite complex, c o m p u t e r simulation techniques are straightf o r w a r d and should provide considerable insight into neural processes if our model is justified.

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5. Applications of the Model

In this section we cite some of the neural functions which, we believe, lend themselves most readily to an interpretation in terms of our model. The discussion is in outline form, suggesting, rather than carrying out, the detailed application of netlet dynamics. Some of these studies are now in progress in our laboratory. The netlet concept, apart from making tractable an otherwise impossible computational task, provides these additional features: the redundancy which arises from the cooperative action of neurons in a pool prevent failure of function resulting from unit failure. Second, the netlet has a richness of dynamical properties quite different from those of single neurons. It is, as Mountcastle called it, an elementary unit of organization, capable of providing a great variety of neural mechanisms, some or all of which may be found operating in the central nervous system. (A) SUBCORTICALPHENOMENA The motoneuron pools in the spinal cord have been studied extensively by physiologists and anatomists. Spinal reflex discharges produced by dorsal root stimulation and monitored at the ventral root, in general show both the fast response of the so-called monosynaptic discharge and the slow, asynchronous multisynaptic discharge (Ruch & Patton, 1966). We may assume that the internuncial neurons (and perhaps also the motoneurons) are synaptically connected among themselves and form, to a fair approximation, what we called netlets. The existence of such multiple pathways was already postulated by Lorente de N6 0938). The multisynaptic discharge may then be considered to reflect the dynamic response of the netlets to brief periods of sustained inputs, the so-called afferent volleys. A striking feature of the netlet dynamics discussed in the preceding section is the powerful control which slow steady inputs can exert on the system's response to brief signals. Refer again to Fig. 4 where steady state activity is plotted against steady excitatory input for a class C netlet. Assume that tr is less than tro, and the activity is at its steady state value, which is very small. A brief stimulation may temporarily raise the activity level but it will rapidly return to the low steady state value. If, on the other hand, the steady input were raised slightly into the region between tro and a~rit, then a brief but sufficiently strong stimulus would elicit a strong and sustained response from the netlet. The threshold stimulus strength is seen to depend very sensitively on the level of the steady input tr when o- lies between fro and trcrlt, and is in fact given by the dotted line in Fig. 4(a). Similar strong control can be exerted on some netlets by inhibitory inputs.

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Such control has been attributed for some time to neural centers located in the brain stem reticular formation. One is dealing there with a structure which shows a great deal of homogeneity, great proliferation of dendritic processes and apparent randomness (Scheibel & Scheibel, 1968). The connectivity is markedly anisotropic, however: dendritic branches are distributed in planes perpendicular to the axis of the oblong shape of the structure. Thus presumably strong communication exists between neurons lying more or less in the same perpendicular plane, giving rise to what has been called the poker chip model of the reticular formation. The effect of the reticular formation in modulating spinal mechanisms, established by Niemer & Magoun (1947) is one of many examples of centrifugal control. Fibers from the reticular formation were found terminating among spinal interneurons (Scheibel & Scheibel, 1968) whose apparent netlet character we mentioned above. The striking depression of certain motor reflexes during sleep may be attributed either to inhibitory control or the cessation of facilitatory bias exerted at the spinal cord by the brain stem centers. The effect of the projections from the reticular formation into the cerebral cortex has been associated with the arousal or activation of cortical centers, and even the activation of emotional behavior (Zanchetti, 1967). Absence of the brain stem's "diffuse facilitatory influence" on the cortex leads to sleep or coma (Moruzzi, 1966). As a last example of the activating action of one neural component upon another, we refer again to the effect of spontaneous activity generated in sensory systems to which Granit (1962) has attributed considerable importance. He quotes, for example, the findings of Adrian (1950), according to which continuous activity i.n the olfactory bulb after being extinguished by anaesthetics may be restarted by olfactory stimulation. This effect suggests a hysteresis of the type discussed in connection with netlets receiving sustained inputs. (B) INFORMATIONPROCESSING IN THE CORTEX We now wish to consider processes in the neocortex, where evidence for the netlet structure is perhaps most pronounced. Sensory netlets such as those observed by Mountcastle or by Hubel & Wiesel may be stimulated into a higher level of sustained equilibrium activity by inputs arriving from the corresponding sense receptors if the netlet parameters are appropriate. Such reverberatory activity has been postulated often as the carrier of short term memory and the necessary prerequisite for laying down a more permanent memory trace.

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A possible mechanism for this process is the gradual change of synaptic efficacy--in our notation an increase in the synaptic coupling coefficients k~i. Such synapticfacilitation was postulated by Hebb (1949), Eccles (1966) and others. There has been some recent experimental evidence (Phillips & Porter, 1969; Bliss, Burns & Uttley, 1968; Gartside, 1968) supporting this theory. If we accept the mechanism of synaptic facilitation, we would expect the most frequently stimulated sensory netlets to become more highly excitable because of their increased coupling coefficients and the resulting decrease in the parameter r/. The very sensitive dependence of netlet characteristics on r/is shown in II. A hierarchy of netlet organization in the visual cortex is suggested by the findings of Hubel & Wiesel (1962). Neurons with simple receptive fields, found primarily in area 17 of the visual cortex, may be thought of as receiving inputs directly from neurons in the lateral geniculate nucleus. On the other hand, the so-called complex fields presumably receive their inputs not from the point detectors but represent combinations of simple fields. As an example of the latter we may take fields constituting of an elongated excitatory region bordered by a halo of inhibition. Such a neuron is thus a detector of a bright linear object. Moreover the line has a specific location in space and a well-defined orientation. As an example of a complex field, a neuron may respond most strongly to a bright line with a definite orientation, but differing from the simple field in that the location may now be immaterial. Neurons with these properties represent a higher level of abstraction. Their function may be explained by postulating afferents from all the simple field cells having the same orientation. We mentioned that the neuronal design implicit in the above findings is apparently genetically determined. The necessity to respond to lines, edges, and even more complex sensory concepts is genetically foreseeable, hence considerable design must exist at the level of the visual cortex. For this reason long term plastic changes in this portion of the brain may be small or even absent altogether. However at a higher level unpredictable combinations of the more primitive gnostic units must be made. We now venture some speculations concerning mechanisms in the association areas. In Fig. 8 we show a schematic of the connectivity matrix of a net in which S represents a primitive sensory area composed of the sensory netlets sl, s2, s3, etc. and A an association area. It is assumed that A also has a netlet structure similar to S. The stippled diagonal blocks in Fig. 8 denote the internal connections of the netlets, the off-diagonal blocks (areas SA and AS) contain the connections leading from one netlet to another. Thus sl, s 2, s3, etc. in the sensory area are seen to have projections to a~, a2, as, etc. respectively in the association area. Also, taking a hint from Sherrington's

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assumption of synaptic facilitation.

reciprocity rule, we assum0 that return fibers exist as shown from al, a2, a3, etc. to sx, s2, s3, etc., respectively. This causes the connectivity matrix to be symmetrical in the large. Of course, the ordering of the netlets in the diagram, as well as the ordering of neurons within each netlet, is completely arbitrary. Part A of the net differs from S in that there is a diffuse background of coupling coefficients over that portion of the connectivity matrix. Another way of saying this is that each of the netlets in A is weakly coupled to every other netlet in A, or that all macroscopic mutual coupling coefficients are non-zero. An anatomical basis for this assumed diffuse connectivity are the so-called U-fibers, axons which are seen to descend from the cortex into the white matter and then re-emerge some distance away. Apart from these U-fibers, the net can be pictured, to a very crude approximation, as a series of two-netlet loops o f the type shown in Fig. 9, where the additional input into netlet sl represents the intermittent sensory input. The structure may T.B,

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easily give rise to reverberatory activity, and since sl represents a netlet with a specific semantic label, the activity set up in the loop s~ ~ a 1 is information preserving. (C) LEARNING

It is assumed that an excitatory coefficient kts becomes reinforced if the post-synaptic potential coincides with the firing of the cell, i.e. if the firing of the j t h neuron contributes to the firing of the ith neuron. The effect is small and cumulative so that it may be neglected when we compute net dynamics over short periods of time. This assumption was called by Caianiello (1961) the adiabatic learning hypothesis. Prolonged activity within a netlet will cause a number of connections to become reinforced. I f this activity shifts ergodically within the netlet, most if not all of its coupling coefficients will in time become reinforced. Thus the short-term memory o f an event, represented by the sustained activity in netlets located in the sensory and association areas, generates the long-term memory consisting of a heightened excitability of the same netlets. Association of different sensory events m a y be explained by the effect of the simultaneous excitation of two or more sensory netlets. This would lead to the reinforcement of the mutual coupling coefficients for the corresponding netlets in the association area. In Fig. 8 we have assumed that events characterized by s~ and s3 have been associated by repeated pairing. In the association area the crosshatched blocks denote reinforced areas. The same situation is diagrammed in Fig. 10. Here the arrows are the macroscopic mutual coupling coefficients between netlets; the dashed arrows are reinforced connections. Without going into the details of the neural dynamics it can be seen that the resuIt o f the changes will be that in the future the presentation of event s 1 alone will lead to some activity in a 3 and s 3. This mechanism is a simple explanation for the so-called sensorysensory cortical conditioning experiments (Morrell, 1957, 1967; Yoshii,

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Shimokichi & Yamaguchi, 1957, 1960). Here stimuli from different modalities are paired and changes in neural activity in the sensory cortical areas are observed. In a recent experiment (Morrell, 1967), neurons in the visual association cortex were monitored and their characteristic responses to an auditory and a visual stimulus were noted. A third response was observed when both stimuli were presented simultaneously. After frequent pairing of the two stimuli, the visual stimulus by itself was able to elicit the response characteristic of the paired stimuli. It should be pointed out that the structuring assumed in areas A, SA and A S (Fig. 8) is not essential to produce the effect described. It can be shown, in fact, that sensory-sensory conditioning is achieved if the sensory netlets s~, s2, s3, etc., project diffusely into an unstructured association net A, as long as A is sufficiently large, so that little or no overlap exists between the sets of neurons in A, directly affected by different sensory netlets. (D) CONDITIONED REFLEXES

By considering also the portion of the cortex which is concerned with motor outputs, we can simulate the perhaps most thoroughly studied cerebral function: the conditioned reflex. Two possible schemes are shown in Fig. 11.

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Two sensory netlets and their respective association netlets are represented by st, s2, at, a2. In Fig. ll(a), a netlet m t in the m o t o r cortex receives projections from a t. The m o t o r action corresponding to m t is thus the ultconditioned reflex to the sensory event st. The same arguments used above in connection with sensory-sensory conditioning now show that frequent pairing of the events st and s2 will eventually cause the conditioned stimulus sz by itself to elicit the m o t o r reaction rn~. The alternative shown in Fig. 1 l(b) where m t is triggered by sl rather than at will produce the same effect. Conditioned reflexes were simulated by an approach similar to the above in a homogeneous association net (Harth & Edgar, 1967). It was shown

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there that such a system is capable of performing a variety of tasks which can be related to known cerebral functions. The model employed was essentially static and considerably more primitive than the present one, but the conclusions reached there are still applicable.

(E) CONCLUSIONS

The CNS is viewed as a network of neurons which combines in its structure genetically determined design and randomness. In the model we have discussed, these features are separated by considering probabilistic nets, or netlets, as building blocks from which larger dynamical structures are assembled. Such a division into randomness in-the-small and design in-thelarge most certainly represents an oversimplification of neuronal connectivity. On the other hand, the many examples quoted, in which neural populations exhibit a functional unity, provide considerable biological basis for this model. The second ingredient of the model was the assumption that the temporal and spatial microstructure of neuronal activity may be neglected and that brain function is determined to a large extent by the fractional number of neurons active in the different netlets. The computations and network simulations carried out in II have revealed some of the richness of dynamical behavior of interacting netlets. A number of the neural mechanisms encountered in this study suggest possible explanations for specific brain functions. It is shown, for example, how slowly varying inputs into a netlet may produce hysteresis effects, thus providing a short-term memory in the form of the level of sustained activity in the netlet. Small changes in afferent excitation may cause the activity to change by several orders of magnitude. This is best illustrated in Fig. 15 of II. The transitions induced are extremely sharp and should be highly reproducible for large netlets. Spontaneous activity, endogenous in a netlet, has an effect very similar to that of sustained excitatory input. Apart from being able to produce hysteresis it may very sensitively affect the excitability of the netlet. The most primitive feature of the interaction between two netlets is the tonic or inhibitory effect of the activity in one netlet on the dynamics of the other. Many interactions of this type appear to exist in the CNS. It is hoped that quantitative comparisons of specific examples with the present model will be possible. Finally the short term memory of netlets discussed above, coupled with the assumption of synaptic facilitation, was shown to provide a mechanism for permanent learning if the interpretation of netlet activity is accepted. The classical conditioned reflex lends itself to a particularly simple explanation in terms of the netlet model.

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