The connectivity of the brain: multi-level quantitative analysis

 We develop a mathematical formalism or calculating connectivity volumes generated by specific topologies with various physical packing strategies. We consider four topologies (full, random, nearest-neighbor, and modular connectivity) and three physical models: (i) interior packing, where neurons and connection fibers are intermixed, (ii) sheeted packing where neurons are located on a sheet with fibers running underneath, and (iii) exterior packing where the neurons are located at the surfaces of a cube or sphere with fibers taking up the internal volume. By extensive cross-referencing of available human neuroanatomical data we produce a consistent set of parameters for the whole brain, the cerebral cortex, and the cerebellar cortex. By comparing these inferred values with those predicted by the expressions, we draw the following general conclusions for the human brain, cortex, and cerebellum: (i) Interior packing is less efficient than exterior packing (in a sphere). (ii) Fully and randomly connected topologies are extremely inefficient. More specifically we find evidence that different topologies and physical packing strategies might be used at different scales. (iii) For the human brain at a macrostructural level, modular topologies on an exterior sphere approach the data most closely. (iv) On a mesostructural level, laminarization and columnarization are evidence of the superior efficiency of organizing the wiring as sheets. (v) Within sheets, microstructures emerge in which interior models are shown to be the most efficient. With regard to interspecies similarities and differences we conjecture (vi) that the remarkable constancy of number of neurons per underlying square millimeter of cortex may be the result of evolution minimizing interneuron distance in grey matter, and (vii) that the topologies that best fit the human brain data should not be assumed to apply to other mammals, such as the mouse for which we show that a random topology may be feasible for the cortex. Received: 14 December 1994/Accepted in revised form: 23 May 1995     

Cortico-thalamic interplay and the security of operation of neural assemblies and temporal chains in the cerebral cortex

 Much evidence suggests that the mammalian thalamus is not merely a set of nuclei relaying signals to the cerebral cortex, but is engaged in two-way interplay with it. Three important features constrain ideas about the nature of this interplay: (i) thalamic projection neurones lack local axon collaterals; (ii) most cortico-thalamic projections have very long axonal conduction time; (iii) in the waking state the membrane potential of thalamic projections cells appears to be poised just beneath threshold for firing. It is proposed that cortico-thalamo-cortical pathways represent connections between different cortical loci which have higher security than the direct cortico-cortical route. Thus each thalamo-cortical projection neurone can have a singular and pivotal role in the activation of one or more cortical neural assemblies. The long delays of cortico-thalamic conduction suggest that the cortico-thalamo-cortical loop also plays a crucial role in the operation of time-structured neural assemblies (‘synfire chains’: Abeles), by providing a high-security link from one active node of the chain to nodes activated at a later time in the sequence. It is predicted that, in the waking animal, thalamic projection cells should have a response selectivity to complex percepts and concepts, conferred on them by the cortical assemblies in whose activation they participate. Received: 30 November 1995/Accepted: 3 June 1996     

On the correlation dimension of optokinetic nystagmus eye movements: computational parameters, filtering, nonstationarity, and surrogate data

 We discuss the estimation of the correlation dimension of optokinetic nystagmus (OKN), a type of reflexive eye movement. Parameters of the time-delay reconstruction of the attractor are investigated, including the number of data points, the time delay, the window duration, and the duration of the signal being analyzed. Adequate values are recommended. Digital low-pass filtering causes the dimension to increase as the filter cutoff frequency decreases, in accord with a previously published prediction. The stationarity of the correlation dimension is examined; the dimension appears to decrease over the course of 120 s of continuous stimulation. Implications for the reliable estimation of the dimension are considered. Several surrogate data sets are constructed, based on both early (0–30 s) and late (100–130 s) OKN segments. Most of the surrogate data sets randomize some aspect of the original OKN, while maintaining other aspects. Dimensions are found for all surrogates and for the original OKN. Evidence is found that is consistent with some amount of deterministic and nonlinear dynamics in OKN. When this structure is randomized in the surrogate, the dimension changes or the dimension algorithm ceases to converge to a finite value. Implications for further analysis and modeling of OKN are discussed. Received: 30 August 1996/Accepted in revised form: 13 November 1996     

Investigation of the dynamics underlying periodic complexes in the EEG

Periodic complexes (PC), occurring lateralised or diffuse, are relatively rare EEG phenomena which reflect acute severe brain disease. The pathophysiology is still incompletely understood. One hypothesis suggested by the alpha rhythm model of Lopes da Silva is that periodic complexes reflect limit cycle dynamics of cortical networks caused by excessive excitatory feedback. We examined this hypothesis by applying a recently developed technique to EEGs displaying periodic complexes and to periodic complexes generated by the model. The technique, non-linear cross prediction, characterises how well a time series can be predicted, and how much amplitude and time asymmetry is present. Amplitude and time asymmetry are indications of non-linearity. In accordance with the model, most EEG channels with PC showed clear evidence of amplitude and time asymmetry, pointing to non-linear dynamics. However, the non-linear predictability of true PC was substantially lower than that of PC generated by the model. Furthermore, no finite value for the correlation dimension could be obtained for the real EEG data, whereas the model time series had a dimension slighter higher than one, consistent with a limit cycle attractor. Thus we can conclude that PC reflect non-linear dynamics, but a limit cycle attractor is too simple an explanation. The possibility of more complex (high dimensional and spatio-temporal) non-linear dynamics should be investigated. Received: 26 February 1998 / Accepted in revised form: 24 August 1998     

Topographic analysis of dimension estimates of EEG and filtered rhythms in epileptic patients with complex partial seizures

Nonlinear dynamic properties were analyzed on the EEG and filtered rhythms recorded from healthy subjects and epileptic patients with complex partial seizures. Estimates of correlation dimensions of control EEG, interictal EEG and ictal EEG were calculated. The values were demonstrated on topograms. The delta (0.5–4 Hz), theta (4–8 Hz), alpha (8–13 Hz), beta (13–30 Hz) and gamma (30–40 Hz) components were obtained and considered as signals from the cortex. Corresponding surrogate data was produced. Firstly, the influence of sampling parameters on the calculation was tested. The dimension estimates of the signals from the frontal, temporal, parietal and occipital regions were computed and compared with the results of surrogate data. In the control subjects, the estimates between the EEG and surrogate data did not differ (P > 0.05). The interictal EEG from the frontal region and occipital region, as well as its theta component from the frontal region, and temporal region, showed obviously low dimensions (P < 0.01). The ictal EEG exhibited significantly low-dimension estimates across the scalp. All filtered rhythms from the temporal region yielded lower results than those of the surrogate data (P < 0.01). The dimension estimates of the EEG and filtered components markedly changed when the neurological state varied. For each neurological state, the dimension estimates were not uniform among the EEG and frequency components. The signal with a different frequency range and in a different neurological state showed a different dimension estimate. Furthermore, the theta and alpha components demonstrated the same estimates not only within each neurological state, but also among the different states. These results indicate that the theta and alpha components may be caused by similar dynamic processes. We conclude that the brain function underlying the ictal EEG has a simple mechanism. Several heterogeneous dynamic systems play important roles in the generation of EEG. Received: 10 December 1999 / Accepted in revised form: 8 May 2000     

Surrogate data analysis of sleep electroencephalograms reveals evidence for nonlinearity

 We tested the hypothesis of whether sleep electroencephalographic (EEG) signals of different time windows (164 s, 82 s, 41 s and 20.5 s) are in accordance with linear stochastic models. For this purpose we analyzed the all-night sleep electroencephalogram of a healthy subject and corresponding Gaussian-rescaled phase randomized surrogates with a battery of five nonlinear measures. The following nonlinear measures were implemented: largest Lyapunov exponent L1, correlation dimension D2, and the Green-Savit measures δ2, δ4 and δ6. The hypothesis of linear stochastic data was rejected with high statistical significance. L1 and D2 yielded the most pronounced effects, while the Green-Savit measures were only partially successful in differentiating EEG epochs from the phase randomized surrogates. For L1 and D2 the efficiency of distinguishing EEG signals from linear stochastic data decreased with shortening of the time window. Altogether, our results indicate that EEG signals exhibit nonlinear elements and cannot completely be described by linear stochastic models. Received: 21 December 1995/Accepted in revised form: 19 March 1996     

Use of the fractal dimension for the analysis of electroencephalographic time series

 Electroencephalogram (EEG) traces corresponding to different physiopathological conditions can be characterized by their fractal dimension, which is a measure of the signal complexity. Generally this dimension is evaluated in the phase space by means of the attractor dimension or other correlated parameters. Nevertheless, to obtain reliable values, long duration intervals are needed and consequently only long-term events can be analysed; also much calculation time is required. To analyse events of brief duration in real-time mode and to apply the results obtained directly in the time domain, thus providing an easier interpretation of fractal dimension behaviour, in this work we optimize and propose a new method for evaluating the fractal dimension. Moreover, we study the robustness of this evaluation in the presence of white or line noises and compare the results with those obtained with conventional spectral methods. The non-linear analysis carried out allows us to investigate relevant EEG events shorter than those detectable by means of other linear and non-linear techniques, thus achieving a better temporal resolution. An interesting link between the spectral distribution and the fractal dimension value is also pointed out. Received: 21 November 1996 / Accepted in revised form: 1 July 1997     

Extending the HKB model of coordinated movement to oscillators with different eigenfrequencies

 We study the dynamics of a system of coupled nonlinear oscillators that has been used to model coordinated human movement behavior. In contrast to earlier work we examine the case where the two component oscillators have different eigenfrequencies. Problems related to the decomposition of a time series (from an experiment) into amplitude and phase are discussed. We show that oscillations at multiples of the main frequency of the oscillator system may occur in the phase and amplitude due to the choice of a coordinate system and how these oscillations can be eliminated. We derive an explicit equation for the dynamics of the relative phase of the oscillator system in phase space that enables a direct comparison between theory and experiment. Received: 30 December 1994/Accepted in revised form: 27 June 1995     

Temporal variability in a system of coupled mitotic timers

 Cell proliferation is considered a periodic process governed by a relaxation timer. The collective behavior of a system composed of three identical relaxation oscillators in numerically studied under the condition that diffusion of the slow mode dominates. We demonstrate: (1) the existence of three periodic regimes with different periods and phase relations and an unsymmetrical, stable steady-state (USSS); (2) the coexistence of in-phase oscillations and USSS; (3) the coexistence of periodic attractors; and (4) the emergence of a two-loop limit cycle coexisting with both in-phase oscillations and a stable steady-state. The qualitative reasons for such a diversitiy and its possible role in the generation of cell cycle variability are discussed. Received: 18 March 1992/Accepted in revised form: 16 April 1994     

A formalisation of the neural assembly concept 1. Constraints on neural assembly size

Hebb proposed the concept of a neural assembly distributed across cortical tissue as a model for representation of information in the cerebral cortex. Later developments of the concept highlight the need for overlapping membership between independent assemblies, and the spread of activity throughout the assembly once it is activated above a critical level (ignition). Formalisation of the neural assembly concept, especially in relation to quantitative data from the real cortex, is at a very early stage. We consider two constraints on neural assembly size: (1) if a neural assembly is too small the fraction of its neurons that need to be active to ignite the whole assembly becomes unrealistically large; (2) if assemblies in a block of cortical tissue become too large then the block becomes ‘unsafe’, that is, unwanted spread from an active assembly to overlapping ones becomes inevitable. We consider variations in three parameters: neuronal firing threshold; connection density; and the total number of assemblies stored in the block of cortical tissue. Given biologically plausible values for these parameters we estimate maximum assembly size compatible with ignitability of individual assemblies, low probability of unwanted spread to overlapping assemblies, and safe operation of the block as a whole. Received: 7 March 1997 / Accepted in revised form: 1 July 1997     

Fractal and nonfractal analysis of cell images: comparison and application to neuronal dendritic arborization

 Neurons of the rat spinal cord were stained using the Golgi impregnation method. Successfully impregnated neurons from laminae II, III, and VI were subjected to fractal and nonfractal analyses. Fractal analysis was performed using length-related techniques. Since an application of fractal methods to the analysis of dendrite arbor structures requires caution, we adopted as appropriate a nonfractal method proposing a generalized power-law model with two main nonfractal parameters: (i) the anfractuosity, characterizing the degree of dendritic deviation from straight lines; and (ii) an estimate of the total length of arbor dendrites. The anfractuosity can distinguish between two sets of drawings where the fractal methods failed. We also redefine some basic concepts of fractal geometry, present the ruler-counting method, and propose a new definition of fractal dimension. Received: 5 February 2002 / Accepted: 25 June 2002 Acknowledgement. We thank Ing. Dejan Ristanović for preparing the computer program used in this study. Correspondence to: D. Ristanović (e-mail: dusan@ristanovic.com, Tel.: +381-11-3615767)     

Do travelling band solutions describe cohesive swarms? An investigation for migratory locusts

. We explore several hypotheses for the swarming behaviour in locusts, with a goal of understanding how swarm cohesion can be maintained by the huge population of insects (up to 10⁹ individuals) over long distances (up to thousands of miles) and long periods of time (over a week). The mathematical models that correspond to such hypotheses are generally partial differential equations that can be analysed for travelling wave solutions. The nature of a swarm (and the fact that it contains a finite number of individuals) mandates that we seek travelling band (pulse) solutions. However, most biologically reasonable models fail to produce such ideal behaviour unless unusual and unrealistic assumptions are made. The failure of such models, general difficulties encountered with similar models of other migratory phenomena, and possible approaches to alleviate these problems are described and discussed. Received: 25 April 1997 / Revised version: 14 September 1997     

Selective visual attention in a neurocomputational model of phase oscillators

In order to understand the dynamic property of covert selective visual attention, which is different from the proposed mechanism of the spotlight metaphor, a two-layered network of phase oscillators was developed. The first layer is related to the hippocampus and controls attention focus formation. The second layer is related to the visual cortex, and each cortical oscillator in it simulates an assembly of cells coding for a particular stimulus in the sense of feature binding. Selective visual attention is interpreted as the result of the emergent synchronization of hippocampus oscillators and a part of cortical oscillators. Numerical experiments are presented to illustrate attention focus formation and attention shifting from one set of stimuli to another. From a neurocomputational point of view, our results demonstrate that attention is an emergent property of the dynamical cell assemblies responding to the whole visual field. Received: 2 January 1998 / Accepted in revised form: 10 November 1998     

Modelling and analysis of time-lags in some basic patterns of cell proliferation

 In this paper, we present a systematic approach for obtaining qualitatively and quantitatively correct mathematical models of some biological phenomena with time-lags. Features of our approach are the development of a hierarchy of related models and the estimation of parameter values, along with their non-linear biases and standard deviations, for sets of experimental data. We demonstrate our method of solving parameter estimation problems for neutral delay differential equations by analyzing some models of cell growth that incorporate a time-lag in the cell division phase. We show that these models are more consistent with certain reported data than the classic exponential growth model. Although the exponential growth model provides estimates of some of the growth characteristics, such as the population-doubling time, the time-lag growth models can additionally provide estimates of: (i) the fraction of cells that are dividing, (ii) the rate of commitment of cells to cell division, (iii) the initial distribution of cells in the cell cycle, and (iv) the degree of synchronization of cells in the (initial) cell population. Received: 15 September 1997/Revised version: 1 April 1998     

New perspectives in brain information processing

Brain cortex activity, as variously recorded by scalp or cortical electrodes in the electroencephalography (EEG) frequency range, probably reflects the basic strategy of brain information processing. Various hypotheses have been advanced to interpret this phenomenon, the most popular of which is that suitable combinations of excitatory and inhibitory neurons behave as assemblies of oscillators susceptible to synchronization and desynchronization. Implicit in this view is the assumption that EEG potentials are epiphenomena of action potentials, which is consistent with the argument that voltage variations in dendritic membranes reproduce the postsynaptic effects of targeting neurons. However, this classic argument does not really fit the discovery that firing synchronization over extended brain areas often appears to be established in about 1 ms, which is a small fraction of any EEG frequency component period. This is in contrast with the fact that all computational models of dynamic systems formed by more or less weakly interacting oscillators of near frequencies take more than one period to reach synchronization. The discovery that the somatodendritic membranes of specialized populations of neurons exhibit intrinsic subthreshold oscillations (ISOs) in the EEG frequency range, together with experimental evidence that short inhibitory stimuli are capable of resetting ISO phases, radically changes the scheme described above and paves the way to a novel view. This paper aims to elucidate the nature of ISO generation mechanisms, to explain the reasons for their reliability in starting and stopping synchronized firing, and to indicate their potential in brain information processing. The need for a repertoire of extraneuronal regulation mechanisms, putatively mediated by astrocytes, is also inferred. Lastly, the importance of ISOs for the brain as a parallel recursive machine is briefly discussed.     

Simple models for excitable and oscillatory neural networks

 Chains of coupled oscillators of simple “rotator” type have been used to model the central pattern generator (CPG) for locomotion in lamprey, among numerous applications in biology and elsewhere. In this paper, motivated by experiments on lamprey CPG with brainstem attached, we investigate a simple oscillator model with internal structure which captures both excitable and bursting dynamics. This model, and that for the coupling functions, is inspired by the Hodgkin–Huxley equations and two-variable simplifications thereof. We analyse pairs of coupled oscillators with both excitatory and inhibitory coupling. We also study traveling wave patterns arising from chains of oscillators, including simulations of “body shapes” generated by a double chain of oscillators providing input to a kinematic musculature model of lamprey.. Received: 25 November 1996 / Revised version: 9 December 1997     

Generalized dimension of the intersection between EEGs

 The generalized dimension defined by [Mandelbrot (1995) J Fourier Anal Appl special J.P. Kahane issue: 409–432] was applied to studying the interrelationship between various parts of human cerebral cortex in different functional conditions. Taking EEG signals from different brain areas as different sets, the generalized dimensions of their intersections were calculated to describe the interrelationship between them. The results showed that the generalized dimensions of intersections in different brain states decreased according to the following order: rest with eyes open, closed, light sleep, and deep sleep. The generalized dimensions of intersections related to the left or right temporal lobe were higher than the others when the subjects was doing mental arithmetic, and there was a decrease when the subjects listened to soft classical music. In addition, it was found that there was a noticeable difference in singular spectra between epileptic patients and normal subjects, irrespective of whether the epileptic patient was experiencing a seizure or not. Received: 3 July 2000 / Accepted in revised form: 30 October 2000     

Neural net classification of REM sleep based on spectral measures as compared to nonlinear measures

 In various studies the implementation of nonlinear and nonconventional measures has significantly improved EEG (electroencephalogram) analyses as compared to using conventional parameters alone. A neural network algorithm well approved in our laboratory for the automatic recognition of rapid eye movement (REM) sleep was investigated in this regard. Originally based on a broad range of spectral power inputs, we additionally supplied the nonlinear measures of the largest Lyapunov exponent and correlation dimension as well as the nonconventional stochastic measures of spectral entropy and entropy of amplitudes. No improvement in the detection of REM sleep could be achieved by the inclusion of the new measures. The accuracy of the classification was significantly worse, however, when supplied with these variables alone. In view of results demonstrating the efficiency of nonconventional measures in EEG analysis, the benefit appears to depend on the nature of the problem. Received: 10 October 2000 / Accepted in revised form: 26 April 2001     

EEG simulation: variation of spectral envelope, pulse synchrony and ≈40 Hz oscillation

 Macroscopic EEG travelling wave phenomena and cortical pulse synchronisation effects are related within a single simple simulation. Non-specific activation acts to control the transfer function of the simulated cortex, and thus determines the relative amplitude of macroscopic EEG waves generated by rhythmic inputs. When concurrent asynchronous excitatory inputs to separate, local, cortical sites are introduced, the simulation reproduces both gamma-band (40 Hz) electrocorticogram (ECoG) activity and synchronous oscillation of action potential pulse density at the separate sites. The gamma-band ECoG and pulse synchrony effects depend on different mechanisms: the former upon local excitatory/inhibitory interactions, and the latter on cortico-cortical interactions. The pattern of synchronous activity depends upon both structural and dynamic aspects of gain, and is sustained by linearised versions of the simulation’s state equations. Dynamic properties of the simulation, which are independent of scale, describe both microscopic and macroscopic phenomena, all in accord with physiological findings. Received: 25 June 1996 / Accepted in revised form: 29 November 1996     

A theory of hippocampal memory based on theta phase precession

 The neural dynamics of the hippocampal network for memory encoding of novel temporal sequences is proposed based on the theta rhythm modulated firing of place cells called theta phase precession. It is hypothesized that theta phase precession is generated at the entorhinal cortex by phase locking between local field theta oscillation and neural oscillators and that the hippocampal closed network with feedforward and backward projections employ theta phase precession to create selectivity in the associative connections needed for temporal sequence storage. Our analyses and computer experiments reveal that the phase precession generated by phase locking instantaneously endows stable phase relations among neural activities in the successively changing neural population. It is concluded that theta phase precession provides biologically plausible dynamics for the memory encoding of novel temporal sequences as episodic events. Received: 18 December 2002 / Accepted: 18 March 2003 / Published online: 20 May 2003 Correspondence to: Y. Yamaguchi (e-mail: yokoy@brain.riken.go.jp, Fax: +81-48-4676938) Acknowledgements. The author would like to express acknowledgement to Drs. McNaughton and Skaggs for their discussion and comment and to Dr. Amari for his continuous encouragement. Further thanks are given to Mr. Haga and Dr. Wu for their discussion and cooperation.