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Evo-devo seeks the genetic and evolutionary basis for the division of the embryo into distinct modules, and for the partly independent development of such modules. I am particularly fascinated about how basic body parts — arms, legs, torso, head — develop along with the plumbing — blood vessels, arteries — that sustains them. From a developmental perspective, it is even more interesting to think about the afferent and efferent neurons and their axonal processes which comprise the peripheral nervous system. How do these get created and routed to location to serve their respective functions?
Sean B. In chimp DNA, about Because humans and chimps diverged from a common ancestor about 6 million years ago, we can assume that one-half of these difference are chimp-specific occurred int the chimp lines and one half human-specific occurred in our lineage. That still leaves 18 million changes in our line since our last common ancestor. My guess would be somewhere on the order of a few thousand. Because human evolution is largely a matter of the evolution of the size, shape and fine-scale anatomy of structures and of timing in development, it is only logical that switch evolution would be important in the evolution of humans as well.
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The continuing mistake is being seduced into believing that simple rules that can generate patterns on a computer screen are the rules that generate patterns in biology. This lesson is hard to internalize both because of our infatuation with our own abilities and because we often forget that it is the environment that shapes all organisms. If we accept that inference in the brain is carried out in a distributed fashion and that memories are encoded in a population of cells, then it seems we need a mechanism whereby ensembles of neurons — possibly at some distance from one another — can be coordinated during learning and inference.
Neuronal oscillations have been suggested as one possible mechanism. Neurons that prefer similar patterns of phase coupling exhibit similar changes in spike rates, whereas neurons with different preferences show divergent responses, providing a basic mechanism to bind different neurons together into coordinated cell assemblies. Waterfall Optical Illusion Stare at a waterfall for 30 seconds then look away at some rocks; the rocks appear to be rising.
Even in the absence of movement there is some spontaneous firing. Thus when you start at the stationary rocks, the downward responsive neurons have been damped and the spontaneous firing of the upward responsive neurons have a temporary advantage since the usual balance of power — both populations spontaneously firing at about the same rate — has shifted.
Changes in the quantity of grey matter neurons, neuropil — dendrites and unmyelinated axons, glial cells and white matter myelinated axons in the period from 6 to 25 years of age. Back to front, unused axon die off coupled with myelination, plus connections between the frontal cerebral cortex the cerebellar cortex.
This cerebral-cerebellar pathway — and evidence of a cerebellar role in language — relieves some of the pressure on the Single Algorithm hypothesis. Although the number of neurons increases, they cannot increase the absolute number of connections each one makes. What tends to happen is that, as absolute brain size increases, the proportional connectivity decreases. Area 10 is involved with memory and planning, cognitive flexibility, abstract thinking, initiating appropriate behavior and inhibiting inappropriate behavior, learning rules, and picking out relevant information from what is perceived through the senses.
Non-primate mammals have two major regions of the prefrontal cortex, and primates have three. This new region is apparently unique to primates and is concerned mainly with the rational aspects of decision making, which are our conscious efforts to reach a decision. This region is densely inteconnecgted with other regions that are larger in human brains — the posterior parietal cortex and temporal lobe cortex — and outside the neocortex, it is connected to several cell groups in the dorsal thalamus that are also disproportionately enlarged, the medial dorsal nucleus and pulvinar.
Primates have more cortical areas than other mammals. It has been found that they have nine or more premotor areas, whereas non-primates have only two to four. It is tempting to think that because we humans are higher functioning, we would have more cortical areas than other primates. Indeed, very recent evidence indicates that unique areas have been found in the visual cortex of the human brain. David Heeger at New York University has just discovered these new areas which hare not found in other primates.
For the most part, however, additional cortical areas have not been found in humans. It appears that other primates, not just the great apes, also have cortical areas that correspond to our language areas and tool-use areas. The planum temporale is larger on the left side that the right side in humans, chimps and rhesus monkeys, but it is microscopically unique in the left hemisphere of the humans. Specifically what is different is that the cortical minicolumns of the planum temporale are larger and the area between the columns is wider on the left of of the human brain than on the right side, while in chimps and rhesus monkeys the columns and the inter-columnar spaces are the same size on both sides of the brain.
Several scientists have suggested that the supragranular layers, and the networks of connections they form between the cortical locations, participate heavily in higher cognitive functions. This is accomplished by linking motor, sensory and association areas. These areas receive sensory inputs from high-order sensory systems, interpret them in the light of similar past experiences, and function in reasoning, judgement, emotions, verbalizing ideas, and storing memory. Cortical neurogenesis can be divided into an early and a late period.
The length of time and number of cell cycles spent in the early period of cell division will ultimately determine the number cortical columns that will be found in any given species, The length of time and the number of cell cycles spent in the later period may determine the number of individual neurons within a cortical column.
A higher number of early divisions will result in a larger cortical sheet and a higher number of later divisions will resulting a higher number of neurons within an individual column. Gazzaniga also cites evidence of differences between human and ape mirror neurons. In monkeys, the presence of the object — in, say, the context of grasping — appears to be necessary to enable motor neuron activation, whereas mimed actions suffice to cause firing in humans. Research on the representation information from the rat whisker pad in the primary somatosensory cortex showed that topographic map formation is highly plastic; the lesson being that the machinery for building representations in primary sensory cortex is adaptively keyed to environmental changes.
You can think of the above lessons as the use cases which I try to keep in mind when I attempt to reverse engineer the brain. In Defense of the Single Algorithm Hypothesis There are certainly plenty of dimensions along which to differentiate cells and circuits belonging to the or so different functional areas in the cortex: the histlogy of individual cells, the cytoarchitecture of cell complexes, neurotransmitters, genetic pathways, connections to thalamus, cerebellum, etc. Even so, none of this precludes the possibility that all of these component areas are running the same algorithm — they may have different implementations and the unifying algorithm may be applied to different data, but these differences are not algorithmic.
The variations that distinguish components may be necessary to orchestrate their simultaneous application to different data thereby allowing parallelism. I have heard it said by a number of computer scientists and computational neuroscientists that the genome is not big enough — does not encode enough information — to completely specify the structure of the cerebral cortex and therefore it is necessary to learn this structure in an unsupervised manner.
There is considerable evidence to suggest that a great deal of the structure of the cortex is determined by complex genomic regulatory networks that use a variety of standard developmental machinery, e. In Defense of the Quantitative Sufficiency Hypothesis The cortical coprocessor model — Suppose that there are no or very few cortical areas that are present in humans but not in our closest primate relatives and that each area is histologically and cytoarchitecturally homogeneous.
Also suppose that the intra-areal and extra-cortico connections in humans are also realized in our closest relatives in kind if not in quantity. In this case it would be relatively simple for additional rounds of cell division during fetal brain development to exploit local chemical gradients and cell-differentiation signals to proportionally expand the cortex.
This is analogous to how semiconductor designers take advantage of new fabrication technologies to improve products without making significant changes to the design. As it becomes feasible to print more transistors on a single die, you can increase the number of processing cores, SIMD lanes, cache sizes, etc. It would be interesting to find a study comparing the maturation of mammalian brains relative to the normalized volume of both white and grey matter. Structural morphological modularity at the genomic level enables computational scaling just as modular circuit designs enable computational scaling in modern computer architectures.
The level of granularity due to algorithmic parsimony if such a principle can be said to apply to the cortex is probably more than a logic gate but how much more so is open to debate. Additional stages of prenatal neurogenesis could plausibly increase the depth of combinatorial neural circuits thus facilitating longer chains of inference and deeper recursive embedding. Informational encapsulation in which specific competences do not have to appeal to other cognitive modules seem rare. Even if high-level cognition is not modular, the neural substrate on which it depends does appear to be highly modular in its construction — in the engineering sense of the structure of the brain being divided into parts that can be developed and operated independently.
Connections from cortical areas implicated in decision making, language, speech, movement execution and planning to motor and sequence machinery in thalamic nuclei and the cerebellar cortex call for a more complicated set of algorithmic principals than I expect Hinton, DiCarlo and Lewicki had in mind. As a species we are biased to think that our cognitive capacities are well beyond the apes we see in the wild or in zoos — as opposed to those few raised in captivity in a rich environment interacting almost constantly with humans.
We tend to exaggerate or overemphasize our innate individual capacities and downplay the benefit we derive from civilization including written and spoken language, various affordances for augmenting our finite short and long term memory, and the readily available sources of knowledge available through books, libraries and now the Internet. It seems reasonable to assume given what we know about the physiology that differences in cognitive capacity between apes and humans are — as Darwin suggested — one of degree and not of kind.
What do you get with deeper combinatorial circuits? Left: Nissl-stained visual cortex of a human adult. They are contained within the middle ear space and serve to transmit sounds from the air to the fluid-filled labyrinth cochlea. The bones that comprise the intermediate links in the hinge are believed to have evolved into the small bones that are part of the mammalian auditory system. The Michigan arrays allow a higher density of sensors for implantation as well as a higher spatial resolution than microwire MEAs.
They also allow signals to be obtained along the length of the shank, rather than just at the ends of the shanks. In contrast to Michigan arrays, Utah arrays are 3-D, consisting of conductive silicon needles. However, in a Utah array signals are only received from the tips of each electrode, which limits the amount of information that can be obtained at one time.
The SNS and PSNS complement one another and work in relative opposition — simplifying a good deal — in the regulation of internal organs and glands. It is, however, constantly active at a basal level to maintain homeostasis. Anatomical interconnections are grouped into four main behavioral compartments. Numbers: Brodmann designations. During translation, protein-constructing ribosomes read the mRNA sequences and translate them into amino acid sequences. Each three-nucleotide codon specifies a particular amino acid.
Once complete the sequence still needs to be folded into a particular conformation in order to serve its purpose in the organism. A Spiking in one area may depend on population activity local field potentials, LFPs occurring in multiple areas. B Many neurons are sensitive to oscillatory LFP activity occurring in particular frequency bands; filtering all LFPs at this frequency and extracting phases can reveal patterns of phase coupling between LFP channels.
D Given novel LFP phases as input, the model generates a predicted coupling-based spike rate output, which can then be compared with the measured spike rate. E The procedure described above can be applied to multiple simultaneously recorded neurons. G Shared variability in coupling-based rates is compactly described by a single phase coupling network that defines a cell assembly. That is, it is possible to identify large-scale patterns of LFP—LFP phase coupling G that explain a significant fraction of the variation in spike rates for a large ensemble of neurons distributed across multiple brain areas.
H Multiple functional ensembles, each spanning several brain areas, overlap in space. I Interference between ensembles is minimized when each assembly responds to a different frequency assemblies A and C or distinct phase-coupling pattern assemblies A and B. J Frequency and pattern selectivity permits dynamic, independent coordination of multiple coactive ensembles. Allometric changes in primary components of telencephalon. The anatomical connection pathways among posterior and anterior neocortex PC, AC , striatum S , and pallidum P are shown for small-brained a and large-brained b mammals.
Sensory inputs vision, audition, touch arrive at thalamus T ; projection loops connect thalamus with cortex and cortex to striatum to pallidum and back to thalamus; both pallidal and motor cortex efferents target brainstem motor nuclei dashed box. The connectome for C. Elegans ; its nervous system consists of non-spiking neurons each one a highly specialized analog computer. A common feature is an interplay between processes of stabilizing selection and processes of relaxed selection at different levels of organism function.
These may play important roles in the many levels of evolutionary process contributing to language. A biological solution to a fundamental distributed computing problem. Science , —, Amunts, A. Schleicher, and K. Cytoarchitecture of the cerebral cortex — more than localization. NeuroImage , 37 4 —, Balsters, E. Cussans, J. Diedrichsen, K. Phillips, T.
Preuss, J. Rilling, and N. Evolution of the cerebellar cortex: The selective expansion of prefrontal-projecting cerebellar lobules. NeuroImage , 49 3 —, Bear, Barry Connors, and Michael Paradiso. Neuroscience: Exploring the Brain Third Edition. A transcriptomic atlas of mouse neocortical layers. Neuron , 71 4 —, Callaerts, G. Halder, and W. PAX-6 in development and evolution.
Annual Review of Neuroscience , —, Oscillatory phase coupling coordinates anatomically dispersed functional cell assemblies. Proceedings of the National Academy of Sciences , 40 —, Endless forms most beautiful: the new science of evo devo and the making of the animal kingdom. Casagrande and G. Bruce Goldstein, editor, Encyclopedia of Perception, Volume 1 , pages — Cyclopia and defective axial patterning in mice lacking sonic hedgehog gene function.
Nature , —, Christensen, A. Estevez, X. Yin, R. Fox, R. Morrison, M. McDonnell, C. In neural mass models, we ignore this possibility because we can only couple the expectations or first moments. There are several devices that are used to compensate for this simplification. This implicitly encodes variability in the postsynaptic depolarisation, relative to the potential at which the neuron would fire. This form of neural mass model has been used extensively to model electrophysiological recordings e.
In summary, neural mass models are special cases of ensemble density models that are furnished by ignoring all but the expectation or mean of the ensemble density.
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This affords a considerable simplification of the dynamics and allows one to focus on the behavior of a large number of ensembles, without having to worry about an explosion in the number of dimensions or differential equations one has to integrate. The final sort of model we will consider is the generalisation of neural mass models that allow for states that are functionals of position on the cortical sheet.
These are referred to as neural field models and are discussed in the following sections. The density dynamics and neural mass models above covered state the attributes of point processes, such as EEG sources, neurons, or neuronal compartments. An important extension of these models speaks to the fact that neuronal dynamics play out on a spatially extended cortical sheet.
This allows one to formulate the dynamics of the expected field in terms of partial differential equations in space and time. These are essentially wave equations that accommodate lateral interactions. Although we consider neural field models last, they were among the first mean-field models of neuronal dynamics  , . Key forms for neural field equations were proposed and analysed by  — . These models were generalized by  ,  who, critically, considered delays in the propagation of spikes over space.
The introduction of propagation delays leads to dynamics that are very reminiscent of those observed empirically. Typically, neural field models can be construed as a spatiotemporal convolution c. The formal similarity with the neural mass model in 37 is self-evident. These sorts of models have been extremely useful in modeling spatiotemporally extended dynamics e.
This approximation is valid when the axonal delays contribute mostly to the dynamics, for instance in large-scale networks, when the local dynamics are much faster than the network dynamics. It is easy to show that most realistic connectivity kernels provide a neural wave equation like Equation 40; this is due to the fact that the connectivity must remain integrable. As above, the parameter c is the propagation velocity of action potentials traveling down an axon. This class of models are also sometimes referred to as continuous attractor neural networks CANN.
Amari also identified criteria to determine if only one bump, multiple bumps, or periodic solutions exist and if they are stable. This simple mathematical model can be extended naturally to accommodate multiple populations and cortical sheets, spike frequency adaptation, neuromodulation, slow ionic currents, and more sophisticated forms of synaptic and dendritic processing as described in the review articles  ,  , . Spatially localized bump solutions are equivalent to persistent activity and have been linked to working memory in prefrontal cortex  , .
During behavioral tasks, this persistent elevated neuronal firing can last for tens of seconds after the stimulus is no longer present. Such persistent activity appears to maintain a representation of the stimulus until the response task is completed. Local recurrent circuitry has received the most attention, but other theoretical mechanisms for the maintenance of persistent activity, including local recurrent synaptic feedback and intrinsic cellular bistability  ,  , have been put forward.
Single bump solutions have been used for neural modeling of the head-direction system  —  , place cells  —  , movement initiation  , and feature selectivity in visual cortex, where bump formation is related to the tuning of a particular neuron's response . Here the neural fields maintain the firing of its neurons to represent any location along a continuous physical dimension such as head direction, spatial location, or spatial view. The mathematical analysis of the neural field models is typically performed with linear stability theory, weakly nonlinear perturbation analysis, and numerical simulations.
With more than one population, nonstationary traveling patterns are also possible. In two dimensions, many other interesting patterns can occur, such as spiral waves  , target waves, and doubly periodic patterns. These latter patterns take the form of stripes and checkerboard-like patterns, and have been linked to drug-induced visual hallucinations . For smooth sigmoidal firing rates, no closed-form spatially localized solutions are known, though much insight into the form of multibump solutions has been obtained using techniques first developed for the study of fourth-order pattern forming systems .
Moreover, in systems with mixed excitatory and inhibitory connectivity or excitatory systems with adaptive currents, solitary traveling pulses are also possible. The bifurcation structure of traveling waves in neural fields can be analysed using a so-called Evans function and has recently been explored in great detail .
Much experimental evidence, supporting the existence of neural fields, has been accumulated see  for a summary. Most of these results are furnished by slice studies of pharmacologically treated tissue, taken from the cortex  —  , hippocampus  , and thalamus . In brain slices, these waves can take the form of synchronous discharges, as seen during epileptic seizures  , and spreading excitation associated with sensory processing .
For traveling waves, the propagation speed depends on the threshold, h , which has been established indirectly in real neural tissue rat cortical slices bathed in the GABA-A blocker picrotoxin by . These experiments exploit the fact that i cortical neurons have long apical dendrites and are easily polarized by an electric field, and ii that epileptiform bursts can be initiated by stimulation. A positive negative electric field applied across the slice increased decreased the speed of wave propagation, consistent with the theoretical predictions of neural field theory, assuming that a positive negative electric field reduces increases the threshold, h , in Equation More and more physiological constraints have been incorporated into neural field models of the type discussed here see Equations 39 and These include features such as separate excitatory and inhibitory neural populations pyramidal cells and interneurons , nonlinear neural responses, synaptic, dendritic, cell-body, and axonal dynamics, and corticothalamic feedback  ,  ,  ,  ,  ,  — .
A key feature of recent models is that they use parameters that are of functional significance for EEG generation and other aspects of brain function; for example, synaptic time constants, amount of neurotransmitter release or reuptake, and the speed of signal propagation along dendrites. Inferences can also be made about the parameters of the nonlinear IF response at the cell body, and about speeds, ranges, and time delays of subsequent axonal propagation, both within the cortex and on extracortical paths e.
It is also possible to estimate quantities that parametrize volume conduction in tissues overlying the cortex, which affect EEG measurements  , or hemodynamic responses that determine the blood oxygen level—dependent BOLD signals . Each of these parameters is constrained by physiological and anatomical measurements, or, in a few cases, by other types of modeling. A key aim in modeling is to strike a balance between having too few parameters to be realistic, and too many for the data to be able to constrain them effectively.
Recent work in this area has resulted in numerous quantitatively verified predictions about brain electrical activity, including EEG time series  ,  ,  , spectra  ,  ,  ,  ,  , coherence and correlations, evoked response potentials ERPs  , and seizure dynamics  ,  , . Inversion of these models has also furnished estimates of underlying physiological parameters and their variations across the brain, in different states of arousal and pathophysiology  ,  , . There are several interesting aspects to these modeling initiatives, which generalize the variants discussed in earlier sections: i synaptic and dendritic dynamics and summation of synaptic inputs to determine potentials at the cell body soma , ii generation of pulses at the axonal hillock, and iii propagation of pulses within and between neural populations.
We now look more closely at these key issues. Assume that the brain contains multiple populations of neurons, indexed by the subscript a , which labels simultaneously the structure in which a given population lies e. Then the spatially continuous soma potential, V a , is the sum of contributions, V ab , arriving as a result of activity at each type of mainly dendritic synapse b , where b indexes both the incoming neural population and the neurotransmitter type of the receptor.
The summation is assumed to be linear, and all potentials are measured relative to the resting potential . For moderate perturbations relative to a steady state, the value of the resting potential can be subsumed into the values of other parameters . As above, the cortex is approximated as a 2-D sheet and r is assumed to be the actual position in the case of the cortex; other structures, such as the thalamus, are linked to the cortex via a primary topographic map. This map links points in a one-to-one manner between structures; i.
Hence, in structures other than the cortex, this dimensional map coordinate, r , denotes a rescaled physical dimension i. The subpotentials, V ab , respond in different ways to incoming spikes, depending on their synaptic dynamics ion-channel kinetics, diffusion in the synaptic cleft, etc. The resulting soma response to a delta-function input at the synapse can be approximated via the differential equation .
If we assume linear propagation, signals propagate as described by the neural field equation Equation By employing population-specific fields and parameters, it allows each population to generate a family of outgoing fields that propagate to different populations in different ways. Critically, the neural field Equation 49 enables very diffuse i. The above equations contain a number of parameters encoding physiology and anatomy e.
In general, these can vary in space, due to differences among brain regions, and in time, due to effects like habituation, facilitation, and adaptation. In brief, time-dependent effects can be included in neural field models by adding dynamical equations for the evolution of the parameters. Typically, these take a form in which parameter changes are driven by firing rates or voltages, with appropriate time constants. The simplest such formulation is  52 where x is the evolving parameter, y is the quantity that drives the evolution, x 0 and y 0 are steady state values, and x 1 is a constant that describes the strength of feedback.
If we use the normalized form 53 then we find the differential equivalent of Equation Here, we first discuss how to find the steady states of neural field models. Important phenomena have been studied by linearizing these models around their steady state solutions. Hence, we discuss linear properties of such models, including how to make predictions of observable quantities from them; including transfer functions, spectra, and correlation and coherence functions.
In doing this, we assume for simplicity that all the model parameters are constant in time and space, although it is possible to relax this assumption at some cost in complexity. Linear predictions from neural field models have accounted successfully for a range of experimental phenomena, as mentioned above.
Nonlinear dynamics of such models have also been discussed in the literature, resulting in successful predictions of epileptic dynamics, for example  ,  , but are not considered here but see the Cognitive and Clinical Applications section.https://gufedivuve.cf/map6.php
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Steady states and global dynamics. Previous work has shown that many properties of neuronal dynamics can be obtained by regarding activity changes as perturbations of a steady state . Spatially uniform steady states can be obtained by solving the preceding equations with all time and space derivatives set to zero, assuming that the parameters are spatially constant. The spatially uniform steady states are thus the solutions of the set of equations 55 which are generally transcendental in form. Linear equations for activity. Of the relevant equations above, all but Equation 48 are linear in Q.
If we Fourier transform the resulting set of linear equations, we find for the fluctuating parts 56,57,58,59,60 where is given by Equation 50 and we have assumed that all the parameters of the equations but not the fields of activity are constant on the timescales of interest. Note that we have assumed the system to be unbounded in order to employ a continuous Fourier transform here.
The case of bounded systems with discrete spatial eigenmodes can be treated analogously. Q j is written as N j to make the distinction between population firing rates and incoming stimulus rates absolutely clear. The element T aj is the response of Q a to a change in N j at the same frequency and wave vector. For example, a scalp potential may involve contributions from several populations, with various weights that may include filtering by volume conduction effects.
Further classes of measurement functions are those relating the neural activity to, for example, local field potentials, multiunit activity, the blood oxygen level—dependent BOLD response that forms the basis of functional magnetic resonance imaging fMRI , the metabolic responses underlying positron emission tomography PET , or single-photon emission computed tomography SPECT. In what follows, we will implicitly absorb M into T for simplicity. Dispersion and stability.
The dispersion relation of linear waves in the system is given by 69 and the system is stable at a particular real k if all the frequency roots of this equation have negative imaginary parts. If the steady state is stable for all k , spectra and other properties of the linear perturbations can be self-consistently defined; otherwise a fully nonlinear analysis is needed. Correlation and coherence functions. In terms of the above expressions, the normalized correlation function and the coherence function, which are both used widely in the literature, are 77,78 respectively.
Time series and evoked potentials. In the case of an impulsive stimulus, the resulting ERP is obtained by setting Case of one long-range population. An important case, in many applications, is the situation where spatial spreading of activity is dominated by the axons of one population, typically because they have the longest range, are most numerous, or have the highest axonal velocity.
The brain's network dynamics depend on the connectivity within individual areas, as well as generic and specific patterns of connectivity among cortical and subcortical areas  ,  , . Intrinsic or intracortical fibers are confined to cortical gray matter in which the cortical neurons reside; these intrinsic connections define the local connectivity within an area. Intracortical fibers are mostly unmyelinated and extend laterally up to 1 cm in the human brain with excitatory and inhibitory connections. Their distribution is mostly invariant under spatial translations homogeneous  ,  , which fits the assumptions on the connectivity function in neural fields so far.
On the other hand, the corticocortical extrinsic fiber system contains fibers which leave the gray matter and connect distant areas up to 20 cm . This fiber system is myelinated, which increases the transmission speed by an order of magnitude, and is not invariant under spatial translations heterogeneous ; in fact it is patchy .
Due to finite transmission speeds, time delays of interareal communication can reach 50— ms  , which is not negligible. Several studies have focused on spatially continuous neural fields, which describe the temporal change of neural activity on local scales, typically within a brain area see  ,  ,  for reviews , assuming homogeneous connectivity and time delays.
As discussed in the previous section, early attempts include neural field theories which approximate the large-scale components of the connectivity matrix as translationally invariant and decaying over space  ,  , . These approaches have been successful in capturing key phenomena of large-scale brain dynamics, including characteristic EEG power spectra  ,  , epilepsy  , and MEG activity during sensorimotor coordination .
Here we review extensions of these efforts and address network stability under variation of i intracortical intrinsic connectivity, ii transmission speed, and iii length of corticocortical extrinsic fibers. All three anatomical attributes undergo characteristic changes during the development of the human brain and its function, as well changing in the aged and diseased brain see  for an overview. These fibers are myelinated and hence to be distinguished from the typically unmyelinated hence slower intracortical fibers. The latter intrinsic fibers have a transmission speed of c hom and a transmission delay.
The adjoint set of spatial biorthogonal basis functions is denoted by.
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It will be generally true except in degenerate cases that only one spatial pattern will become unstable first. Also, and but for simplicity, we drop the tilde in from now on. Let us pause for a moment and reflect upon the significance of Equation In other words, Equation 85 identifies quantitatively how a particular neural activation is impacted by its local and global connectivity in a biologically realistic environment, including signal exchange with finite and varying intracortical versus corticocortical transmission speeds.
Every treatment of the interplay of anatomical connectivity local and global connections and functional connectivity network dynamics will have to be represented in the form of Equation 85 or a variation thereof. In this sense, we have here achieved our goal stated in the introduction of this section. To illustrate the effects of interplay between anatomical and functional connectivity, we discuss a simple example following  , . Then we have an architecture as shown in Figure 1.
The intracortical connections are illustrated as densely connected fibers in the upper sheet and define the homogeneous connectivity W hom. A single fiber connects the two distant regimes A and B and contributes to the heterogeneous connectivity, W het , whereas regime C has only homogeneous connections. Figure 2 shows various connectivity kernels, W hom , that are often found in the literature. Purely excitatory connectivity is plotted in A ; purely inhibitory in B ; center-on, surround-off in C ; and center-off, surround-on in D. The connectivity kernel in C is the most widely used in computational neuroscience.
Qubbaj and Jirsa  discussed the properties of the characteristic Equation 86 in detail, considering separately the special cases of symmetric and asymmetric connectivity, W. Recall that c and c hom are the conduction velocities along extrinsic and intrinsic axons, respectively. The general result of  can be represented as a critical surface separating stable from unstable regimes as shown in Figure 3.
Within the cylindrical component of the surface, the equilibrium of the system remains always stable for all values of c , c hom , and hence a time delay shows no effect. The largest stability region is that for a purely inhibitory kernel followed by that of a local inhibitory and lateral excitatory kernel. The next largest involves a local excitatory and lateral inhibitory kernel. The smallest stability region is obtained for a purely excitatory kernel. The critical surface, at which the equilibrium state undergoes an instability, is plotted as a function of the real and imaginary part of the eigenvalue of its connectivity, W.
Regimes below the surface indicate stability, above instability. The vertical axis shows the time delay via transmission along the heterogeneous fiber. A surprising result is that all changes of the extrinsic pathways have the same qualitative effect on the stability of the network, independent of the local intrinsic architecture. This is not trivial, since despite the fact that extrinsic pathways are always excitatory the net effect on the network dynamics could have been inhibitory, if the local architecture is dominated by inhibition.
Hence qualitatively different results on the total stability could have been expected. Such is not the case, as we have shown here. Obviously the local architecture has quantitative effects on the overall network stability, but not qualitatively differentiated effects. Purely inhibitory local architectures are most stable, purely excitatory architectures are the least stable. The biologically realistic and interesting architectures, with mixed excitatory and inhibitory contributions, play an intermediate role.
When the stability of the network's fixed point solution is lost, this loss may occur through an oscillatory instability or a nonoscillatory solution. The loss of stability for the nonoscillatory solution is never affected by the transmission speeds, a direct physical consequence of its zero frequency allowing time for all parts of the system to evolve in unison. The only route to a non-oscillatory instability is through the increase of the heterogeneous connection strength. For oscillatory instabilities, the situation is completely different.
An increase of heterogeneous transmission speeds always causes a stabilization of the global network state. These results are summarized in Figure 4. Top The relative size of stability area for different connectivity kernels. Bottom Illustration of change of stability as a function of various factors. Gradient within the arrows indicates the increase of the parameter indicated by each arrow. The direction of the arrow refers to the effect of the related factor on the stability change.
The bold line separating stable and unstable regions indicates the course of the critical surface as the time delay changes. This section illustrates neuronal ensemble activity at microscopic, mesoscopic, and macroscopic spatial scales through numeric simulations. Our objective is to highlight some of the key notions of ensemble dynamics and to illustrate relationships between dynamics at different spatial scales. To illustrate ensemble dynamics from first principles, we directly simulate a network of coupled neurons which obey deterministic evolution rules and receive both stochastic and deterministic inputs.
The system is constructed to embody, at a microscopic level, the response of the olfactory bulb to sensory inputs, as originally formulated by Freeman  ,  — . Specifically, in the absence of a sensory input, neurons fire sporadically due to background stochastic inputs. The presence of additional synaptic currents due to a sensory input e. Note that in this section we simulate dynamics at the scale of coupled individual neurons. We can derive directly the predicted ensemble mean response by simply summing over all neurons. We compare this with an explicit model of neural mass dynamics at the mesoscopic scale in the subsequent section.
The planar reduction has slow potassium channel kinetics but fast sodium channels, whose states vary directly with transmembrane potential . Synaptic currents are modeled, for the present purposes, to arise from three sources, The first term represents recurrent feedback from neurons within the ensemble due to their own firing. The coupling term, H c , incorporates both the nature of the all-to-all within-ensemble coupling and the EPSP with parametric strength c. For the present purposes, the EPSP consists of a brief steady current whenever the presynaptic neuron is depolarized.
The external currents, I noise , introduce stochastic inputs e. The final term, I sensory , models sensory input, consisting of a constant synaptic current to a subset of neurons, whenever the sensory stimulus is present. Hence this system permits an exploration of the relative impact of the flow deterministic and diffusive stochastic effects as embodied at the ensemble level by the Fokker-Planck equation Equation 20 at the neuronal network level. The Nernst potentials, conductances, and background current are set so that, in the absence of noise and sensory inputs, each neuron rests just below a saddle-node bifurcation to a limit cycle .
This implies that neurons are spontaneously at rest quiescent but depolarize with a small perturbation. If the perturbation is due to a stochastic train, then the neuron fires randomly at an average rate proportional to the stochastic inputs. However, following a small increase in the constant flow term, due to a sensory input, I sensory , the quiescent state becomes unstable and the neuron evolves on a noise-modulated limit cycle. Figure 5 shows a stochastically driven neuron A compared to a noise-modulated periodic neuron B.
In the former case, the activity is dominated by the stochastic terms. In the latter case, the limit cycle dynamics dominate, although the stochastic inputs modulate the depolarization amplitude. A Stochastically perturbed fixed point. B Limit cycle attractor. As constructed, the effect of the input is to effect a bifurcation in each neuron from stochastic to limit cycle dynamics.
The secondary effect of the appearance of limit cycle dynamics is to suppress the impact of the spatially uncorrelated stochastic inputs. Hence the neurons show an evolution towards phase locking, which was not present prior to the stimulus. As evident in Figure 6B , the increased firing synchrony leads in turn to a marked increase in the simulated local field potentials as individual neurons begin to contribute concurrent ion currents. Once the stimulus ends, there is a brief quiescent phase because all of the neurons have just fired and require a short train of stochastic inputs before they commence firing again.
Interestingly, there is evidence of damped mean-field oscillations in the ensemble following stimulus termination, abating after some further ms. To underscore the observation that the mean synaptic currents evidence an emergent phenomenon, and not merely the super-position of a bursting neuron, the time series of a single neuron is provided in Figure 6C.
Clearly no burst is evident at this scale.
A Raster plot. B Mean synaptic currents. C Time series of a single neuron. The effect of the input is to effect a bifurcation in each neuron from stochastic to limit cycle dynamics phase locking , suppressing the impact of the spatially uncorrelated stochastic inputs.
As evident in A , the increased firing synchrony leads in turn to a marked increase in the simulated local field potentials. The mean synaptic currents evidence an emergent phenomenon, and not merely the superposition of a bursting neuron, as can be seen in C : clearly no burst is evident at this scale.
The impact of the stimulus input on the density of the ensemble is shown in Figure 7 , which shows the spike-timing difference of all neurons in the ensemble with respect to a randomly chosen seed-neuron. The mean spike-timing difference is 0 ms throughout the simulation. This is because the system has complete symmetry, so that all neurons fire, on average, symmetrically before or after any other neuron.
However, as evident in Figure 7A , the variance in relative spike-timing decreases dramatically during the stimulus interval. Of note is that the ensemble variance does not simply step down with the onset of the stimulus, but rather dynamically diminishes throughout the presence of the stimulus. When this occurs, the mean-field term continues to increase in amplitude. Figure 7B shows the evolution of the kurtosis normalized so that a Gaussian distribution has a kurtosis of zero. Prior to the stimulus, and reflecting the weak network coupling, the ensemble has a mesokurtotic broad distribution.
It increases markedly following the stimulus onset, implying a dynamical evolution towards a leptokurtotic peaked distribution. That is, although the parameter values are static, the ensemble mean, variance, and kurtosis evolve dynamically in an inter-related fashion. Hence this system exhibits time-dependent interdependence between its first, second, and fourth moments. This is the sort of coupling between moments of the ensemble density that neural mass models do not capture. A seed neuron is chosen at random and the interneuron spike difference for all other neurons is plotted each time it spikes.
B The normalized fourth moment excess kurtosis derived from a moving frame. It is important to note that the spatiotemporal structure of the noise remains constant throughout the simulation, as does the intra-ensemble coupling. Hence the appearance of phase locking is an emergent feature of the dynamics and has not been imposed. A dynamic contraction of the ensemble cloud occurs whether the pre-existing noise continues unchanged during the stimulus input—hence increasing the firing rate of each neuron—or diminishes so that, in the absence of coupling, firing rates are kept the same on average.
In the latter case as in Figure 5 , there is simply a change from stochastic to periodic firing. The ensemble cloud is visualized directly in Figure 8. The upper row shows the first return map for the ensemble over five consecutive time steps. Six such first return state-space values are plotted for all neurons. To control for changes in spike rate, these plots are normalized to the average firing rate. Values for the seed neuron used in Figure 7 are plotted in red.
The left column shows the ensemble state, prior to the stimulus current. The right column shows the intra-stimulus activity. The contraction of the ensemble is seen clearly. In addition, the first return map confirms that individual neurons have stochastic dynamics prior to the stimulus, which change to periodic i.
The lower row of Figure 7 shows corresponding probability distributions of the inter-neuron spike-timing differences. This reiterates that not only does the distribution contract, but as the mean-field dynamics become strongly nonlinear, the ensemble kurtosis increases markedly from sub- to super-Gaussian. The right column shows the intrastimulus activity. Top row: First return map for the cloud interspike delay over five consecutive time steps, before A and following B synaptic input. The plots are normalized to the average firing rate to control for changes in spike rate.
Lower row C,D : the corresponding spike timing histograms. The ensemble kurtosis increases markedly from sub- to super-Gaussian. As discussed in The Mean-Field Model section, it is possible to study a reduced model representing only the mean ensemble dynamics. This is essentially achieved by generalizing parameter values such as ion channel thresholds from individual point values to population likelihood values.
Freeman  additionally introduced synaptic effects through convolving the inputs with a suitable response kernel as presented in Equation For the simple illustration here, we do not introduce synaptic filtering. For the present purpose, we simulate a single mass with both excitatory and inhibitory neurons  , .
In the microscopic system considered above, interneuron coupling was via a direct pulse during presynaptic depolarization. The dynamics are thus of the form 89 90 where the function G represents the coupling between mean firing rates and induced synaptic currents. Note that both populations receive stochastic inputs but only the excitatory population receives the sensory input I sensory. The functions f ion are the same as for the microscopic system including the slow potassium channel—although they are now parameterized by population-wide estimates.
Figure 9 shows the response of a single neural mass to sensory evoked synaptic currents with the same temporal timing as for the microscopic system. Prior to the stimulus, the system is in a stable fixed point regimen. The stochastic inputs act as perturbations around this point, giving the time series a noisy appearance, consistent with the prestimulus microscopic ensemble activity.
However, the mechanisms are quite distinct: Individual neurons within the microscopic ensemble fired stochastically, but at uncorrelated times. Hence, at the level of the ensemble, such individual events contribute in a piecemeal fashion.
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That is, although individual neurons exhibit nonlinear dynamics, the ensemble mean dynamics are linearly stable to the stochastic inputs until the background current is increased. In the mesoscopic case, the system as a whole is stable to small perturbations prior to the stimulus current. The temporally uncorrelated stochastic inputs are effectively filtered by the response properties of the system around this fixed point to yield the simulated activity.
The stochastic inputs act as perturbations around this point. Although individual neurons exhibit nonlinear dynamics, the ensemble mean dynamics are linearly stable to the stochastic inputs until the background current is increased. Then the fixed point state is rendered unstable by the stimulus current and large amplitude oscillations occur.
These cease following stimulus termination. In the mesoscopic neural mass, the fixed point state is rendered unstable by the stimulus current and large amplitude oscillations occur. This accords with the appearance of stimulus-evoked nonlinear oscillations in the ensemble-averaged response of the microscopic system.
In both models, such oscillations abate following stimulus termination. Hence, at a first pass, this neural mass model captures the mean-field response of the microscopic ensemble to a simulated sensory stimulus. What is lost in the neural mass model? In this model, activity transits quickly from a noise-perturbed fixed point to large amplitude nonlinear oscillations. A brief, rapid periodic transient is evident at the stimulus onset 1, ms. The system subsequently remains in the same dynamic state until the stimulus termination.
This hence fails to capture some of the cardinal properties of the microscopic ensemble, namely the coupling between the first and second moments mean and variance. As discussed above, this process underscores the dynamical growth in the mean-field oscillations and the interdependent contraction of the interneuron spike timing variance shown in Figures 6 and 7.
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Because of this process the system is far more synchronized than prior to the stimulus. What is gained in the neural mass model? The addition of a third dimension i. Hence the flow terms in the neural mass model contribute to the expression of aperiodic dynamics in addition to the stochastic inputs. This is not possible in the planar single neural dynamics of the microscopic system because chaotic dynamics require at least three degrees of freedom. Thus the dimension reduction afforded by the neural mass approximation allows the introduction of more complex intrinsic dynamics, permitting dynamical chaos.
Whilst additional dimensions could be added to the microscopic neurons, this would add to an already significant computational burden. The massive reduction in the computational load of the neural mass approximation also allows extension of the spatial scale of the model by an array of neural masses, coupled to form a small patch of cortical tissue. Such a mesoscopic system can be endowed with additional structure, such as hierarchical  , scale-free  , multiscale  , or small world  properties.
For the present purposes, we couple a single input neural mass, as modeled above, hierarchically to a sheet with internal hyperbolic i. Intersystem coupling is purely excitatory-to-excitatory. As above, synaptic currents are induced by the pulse density of the presynaptic neurons, rather than directly via individual presynaptic depolarization.
The sensory node receives the only direct stimulus-induced currents, The hierarchical nature of the system is embodied by the targeted nature of the sensory inputs and the separate parameterization of parameters that couple masses to or within the sheet, C sens and C sheet , respectively. It would also be possible to increase the degree of forward and backward asymmetry by incorporating purely AMPA-like kinetics for the former and NMDA-like kinetics for the latter, as has been proposed as a mechanism for perceptual inference  , .
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