We address three aspects of neuroscience, each for long being largely overshadowed by the neuron doctrine's hegemony [1]: neuroglia (including here also the systems of neuronal and glia gap junctions), extracellular fluid in neural tissue (“brain cell microenvironment” [2]), and neuromodulatory processes. Our leading notion is that the functional state of neurons, individually and in assemblies, is determined by a set of variables (ion conductances and membrane currents, thresholds for neural discharges, synaptic potentials, ion channel kinetics, etc.), whose values at any one time are to varying degrees affected by interactions and interdependencies of these three components, locally as well as globally, and at largely different time scales. In Section 25.2, we review the essential aspects of each of these components separately. This is to provide the basis for our principal objective to analyze in the Section 25.3 the global dynamics of the complex system these components jointly compose, covering a wide range of temporal scales that is characteristic of multifractals. Accordingly, self-similarity and the absence of any specific time scale ensure instant and automatic adaptation to neural impulse traffic over a wide range of frequencies.
In the normal brain neurons, astrocytes and oligodendrocytes are the most abundant and active cells. They express pannexins and connexins, with protein subunits of two families forming membrane channels. Most available evidence indicates that in mammals endogenously expressed pannexins form only hemichannels, whereas connexins form both gap junction channels and hemichannels. Whereas gap junction channels connect the cytoplasm of contracting cells and coordinate electric and metabolic activity, hemichannels communicate the intra- and extracellular compartments and serve as a diffusional pathway for ions and small molecules. Importantly, continuous glial activation triggered by low levels of anomalous proteins expressed in several neurodegenerative diseases resembles those of acute inflammatory responses triggered by ischemia or infectious diseases [3].
Diffusive coupling by gap junctions between various interneuron types and neurogliaform cells in neocortex is now firmly established [4], as is their virtually boundless distribution [5]. Simulation studies have determined their role for supporting synchronous oscillations [6] [7], and identified complementary interrelations with chemical synapses in interneuronal networks [8]. Electrical coupling between axons is also amply documented [9], providing the opportunity for fast and efficient transfer of action potentials for generating highly coherent output pathways of neuronal networks.
It is not for the lack of compassionate reminders that glia did for a long time not get its proper due in the study of neural systems. Basing his argument on C.J. Herrick's electron micrographs, showing glia to attach itself to and surround nerve soma, axon, and dendrites, Galambos [10] persistently sought to persuade the neuroscience community to think of glia as imposing organization and structure on neurons for processing organized, coherent messages. Forty-five years later, evidence had accumulated to support Galambos' [11] claim. The observations cited in the following are merely a distillation of the multiplicity of the reported data on glia–neuronal interactions, intended to convey the richness and diversity of relevant observations.
Neurochemists generated an avalanche of data, promoting astrocytes [12], one of the members of the macroglia family, to full partnership with pre- and postsynaptic neurons in the “Tripartite Synapse” [13]. This has become a fertile concept for characterizing the complex and reciprocal patterns of interactions between astrocytes and neurons (reviewed by Araque and Navarette [14] and Halassa and Haydon [15]). The dynamics of these interactions is sustained by the astrocytes expressing receptors for virtually all important neurotransmitters [16], providing multiple opportunities for engaging neuron–astrocyte complexes at many target points [17–19]. A prevalent feature of astrocyte's role is their participation in the dynamics of calcium in extracellular space on two different space and time scales. For local, short-term interaction, astrocytes (although not electrically excitable) respond to glutamate liberated at presynaptic junctions with calcium spikes [20] which, in turn, release additional glutamate and ATP to neighboring neurons [21] for integrating coincident activity from different dendrites in the same tissue volume [22] [23]. Furthermore, activity-related changes of calcium levels within astrocytes contribute to mobilizing various transmitters and transmission-related substances [24]. Globally and on longer time scales, intercellular propagation of calcium waves [25–27] can support long-range signaling [28] [29]. Recent evidence from the family of connexins suggest that the astrocyte system constitutes a network of communicating cells with definite spatial organization [30] where intercellular communication is controlled by endogenous signals [31].
The dynamics of neuron–glia interaction is complicated by two circumstances: first, due to activity-dependent morphological changes of astroglial processes ending at synaptic regions [32–34]; and, the second, due to a complex anatomical organization of spatial nonoverlapping domains with limited interdigitation of processes from adjacent cells [35–37]. Each domain encompasses some 2 million synapses in human brain [38] as an area of the neuropil that is controlled by a single astrocyte. Moreover, parts of this territory can be controlled autonomously by specialized astrocyte microdomains of filopodia with distinct motility [39]. Groups of neurons are also enwrapped by a layer of lattice-like material: this perineuronal net forms stable complexes surrounding synapses [40] [41], seemingly affecting short-term synaptic plasticity [42].
On a modular basis, computational simulations of different manifestations of astrocyte–neuron interactions contribute to gaining some insight into their functions Astrocytes enwrapping synapses were shown to tune synaptic transmission to optimal efficiency [43]. Patterns of propagated calcium activation in astrocyte networks were modeled by Postnov et al. [44]. De Pita et al. [45] simulated local and global influences of astrocytes on short-term plasticity. Volman et al. [46] developed a biophysical model for the coupling between synaptic transmission and local calcium concentration on an enveloping astrocyte domain, and Goldberg et al.'s [47] simulation showed that nonlinear gap junctions can support long-distance propagation of calcium waves in astrocyte networks. Simulated astrocyte networks display attractor dynamics and bifurcations in phase space [48]. On the basis of theoretical arguments, Mitterauer [49] attributed a structural organization in the form of logical functions to astrocyte domain organization, suggesting its role in the economy of normal and pathological brain functions [50] [51]. From a global perspective, Allegrini et al. [52] examined the relation between neurons and astrocytes as a system of two interacting networks. This coarse-grained approach enabled identifying an augmenting effect of an astrocyte field on the computational complexity of a (simulated) neural network. A mean-field model of a neural-glial mass established quantitatively a strong and activity-dependent influence of astrocyte activity on synaptic transmission, and on the synchronization ratio between neurons, confirming the experimentally based conjectures of Sotero and Martinez-Cancino [53].
Extracellular fluid's coming of age is closely associated with the work of Kuffler and Nicholls [54], which identified the diffusion of ions and various neuroactive substances in intercellular clefts of neural tissue. This theme was again taken up by Vizi and Labos [55], documenting nonsynaptic interaction in nervous tissue, subsequently discussed in detail by Agnati et al. [56] and Zoli and Agnati [57], suggesting to view the intercellular communication among cells in the nervous system in two complementary reference frames, one as “wiring” transmission and the other as “volume” transmission: the former being transmission of excitation between synaptically connected neurons, and the latter attributing diffusive distribution of various ions, neuropeptides, and neurotransmitters to extracellular fluid surrounding neurons. A wealth of experimental data, notably with the effects of ion accumulation in the extracellular fluid following tetanic nerve activity, corroborated this conjecture [58] [59]. In a computational model of a reaction–diffusion system, Werner [60] demonstrated that tetanic stimulation of a group of neurons causes extracellular accumulation of calcium ions, which induces spreading activity patterns in surrounding unstimulated neurons. More recently, Froehlich [61] showed that diffusive modulation of extracellular potassium concentration induces state transitions in neurons with distinct changes in oscillatory patterns. Changes of diffusive coupling in neural networks can change normal activity patterns and precipitate pathological ones [62] [63]. The relevance of nonsynaptic diffusion neurotransmission was further extended and refined by Bach-Y-Rita [64]. However, diffusion of neuroactive substances is slowed down by geometric tortuosity and viscosity of macromolecules in the extracelluar space [65] [66]. By changing their geometric shape, dendritic spines can dynamically regulate diffusion in their vicinity [67].
Concerning glutamate, the important excitatory transmitter, disrupted regulation of extracellular glutamate in the central nervous system contributes to and can exacerbate the acute pathophysiology of traumatic brain injury [68]. Glutamate is normally released into the extracellular space by both neuronal and nonneuronal sources [69]. Glutamate uptake is driven by electrochemical gradients across neurons and glia with most (about 90%) glutamate uptake being performed by two glial transporters in the striatum, GLAST and GLT-1 [70].
The notion of neuromodulation originated with a diversity of observations that could not be accounted for within the established principles of the synaptic transmission with transmitter substances exclusively acting locally at synaptic sites [71] [72]. For the present purpose, we suggest reserving the term “neuromodulation” to designate the composite system of all processes affecting synaptic transmission as distinct from individual neuromodulatory processes in the narrower sense, as listed in the following. The foundational insights into the operational principles of neuromodulation originated with the investigation of the polyvalence of neural network functions in crustaceans. A succession of comprehensive reviews by Getting [73], Harris-Warrick and Marder [74], and Marder and Calabrese [75] summarize the repertoire of neuronal functions attributable to the modulating action of extrasynaptic processes on synaptic transmission. They include effects on synaptic efficacy and presynaptic transmitter release, intrinsic neuronal properties, changes of network connectivity, coupling of neural oscillators and filtering sensory input, and spike-time-dependent plasticity [76]. All known neurotransmitter substances are involved in these effects, as are a multitude of peptides [77] [78].
Reports of neuromodulatory effects in higher functions of vertebrates followed the crustacean work in rapid sequence: Hasselmo and associates produced evidence for forebrain cholinergic neuromodulation of cognition (for a review, see [79]). Ascending brain stem neuromodulatory systems (NMSs) of vertebrate brains came to be implicated in learning mechanisms [80], in adaptive behavior [81], and in emotional control processes (for a recent example, see [82]). Central pattern generators are subject to neuron modulation in vertebrates as they are in invertebrates [83]. In totality, the accumulated observational evidence mandates expanding the classical view of a relatively static neuronal “wiring diagram” to a dynamic system subject to ongoing tuning and reconfiguring by a biochemical network of modulators, effective over a wide range of temporal and spatial scales [84]. Combining experimental observations with computational simulations reveals the combinatorial richness of the modulatory network for generating functionally appropriate and adaptive behavior [85–87].
The multitude of phenomena described as neuromodulation fall into two fundamentally different categories [88]. Intrinsic neuromodulation is the condition of the modulator being released by some of the same neurons that are also part of the circuit they modulate [89]. Hansson and Ronnback [90] review several instances of intrinsic modulation of synaptic transmission by astrocytes, related to release and uptake of glutamine at synaptic sites. Events at the “tripartite synapse” fall also in this category. Extrinsic modulation, on the other hand, consists of the activity of functionally distinct systems processes outside of and parallel to the actual synaptic activity, relying on the storage and transport of neuroactive substances in the extracellular fluid compartment. Most of the effects of neuroglia must be attributed to this category. Beyond regulating merely one synaptic region, extrinsic modulation can globally organize ensembles of circuits, and usually works at a time course up to several seconds rather than the milliseconds of synaptic actions of intrinsic modulation. In general, functional neuron ensembles subserving specific functions are multiply modulated, with the modulators themselves being subject to modulation by neurotransmitters stabilized by feedback regulation. The range and variety of this complex dynamic behavior can in principle be appreciated in computational models where changing the intrinsic properties of synapses and neurons and the communication pattern among them can replicate some of the observed phenomena, including state-dependent effects and bistability of neural networks, their moment-to-moment reconfigurations, conditional bursting, oscillations and synchronization [91], and the tuning of sensory pathways [92].
In the following discussion, we will refer to the totality of the interacting complex of glia, extracellular fluid, and the processes of neuromodulation as the NMS. The observations sketched in the Section 25.2 suggest viewing NMS as composed of numerous semiautonomous modules in nonlinear interactions. If it was not already intuitively obvious from the mere inspection of the wealth of diverse observations, a definite characterization of (NMS) as a nonlinear complex dynamic system (in the technical sense, see for instance [93]) follows from the revealing observations, reported by Marder [94] and based on numerous earlier studies, that NMS can be implicated in generating many different patterns of neural activity for identical tasks conditions. In other words, the NMS can deliver multiple and different solutions to a given demand or task. This phenomenon qualifies as Multiple Realization which is considered a universal attribute of nonlinear complex dynamical systems [95] [96].
For formulating ideas about NMS, it must of course not be overlooked that neurons themselves (individually and as assemblies) are integral participants, active by contributing to the flux of neuroactive substances in extracellular space, and passive by being affected by them. From moment-to-moment, NMS contributes to determining the neuronal parameters which by themselves constitute a smooth manifold over which they can vary for constraining compensatory combinations [97]. Since our aim is to characterize the function of NMS in its totality at a global level, we take a coarse-grained, nonreductive perspective. This sets our approach apart from studies of stochastic synaptic processes at the molecular level [98] and the multiscale analysis of molecular processes at cellular levels (for a recent overview: see Holcman [99]).
Depending on circumstances and objectives, several methods are typically applied to characterize complex biological reaction networks. For instance, Ederer et al. [100] recommend dividing the network into hierarchically structured modules, each amenable to simulation by ordinary differential equations. Piecewise-linear models are common practice for qualitative simulation of genetic regulatory networks [101]. Lima et al. rely on symbolic dynamics application to bifurcation analysis, and Galatolo et al. [102] advocate computing the long-time statistical limiting behavior and invariant measures. Cooperative complexification and adaptation to dynamic environments can originate from self-organization [103]. Phenomenological models of statistical habituation can be successfully based on linear response theory [104]. Recommending an engineering approach to metabolism, transcription, and translation signaling, Doyle and Csete [105] envision an organization framework of “highly optimized tolerance” (HOT) for ensuring structural stability and robustness of complex biological systems whose component processes involve a wide range of temporal scales.
Here, we adopt an alternative approach that aims at characterizing the global dynamics of NMS in its totality as it emerges from the interaction of its semi-autonomous modules. We make the biologically plausible assumption that each process in the chain of neuromodulatory events can be considered a chemical rate process with exponential decay. Relaxation rates vary over at least a 1000-fold range: from milliseconds at the liberation of transmitter substances at intrinsic modulation, to many seconds of chemically mediated astrocyte network reconfigurations and propagating calcium waves, with the numerous extrinsic modulatory processes exhibiting intermediate rates. This situation invites applying the observation of Hausdorff and Peng [106] that systems presenting time series with widely differing scales of component regulatory mechanisms summate to a system's power law (1/f) scaling, suggestive of its fractal character. Although there is no definitive mathematical proof presently available that time-scale-free functions emerge from superposition of independent relaxation processes, there exist a range of physical mechanisms that do in fact show such micro- to macroscopic conversion, generally in the context of fractal time series [107]. Moreover, numerical analyses of Montroll and Shlesinger [108] established that macroscopic scale-free functions emerge, provided the independent microscopic relaxation processes are of sufficiently large variance, as they are in NMS. This principle was subsequently applied by Anderson [109] to ascertain the power law dependency as an emergent property of systems that contain several exponentially decaying traces and was further extended by Fusi et al. [110] and Drew and Abbott [111] to include cascading exponential processes, the latter for sensory adaptation. However, NMS contains too many unknown rate constants to attempt numerical simulation and determination of a power law exponent. Thus, we need to confine the discussion to the exposition of plausible principles and analogies.
Placing the function of NMS into the domain of fractal time series allows gaining significant insights into its dynamic properties. In the first place, it identifies its scale-invariance as the property of relating the elements of its behavior in time across multiple time scales. This is a characteristic empirical feature of a large number of complex physiological phenomena [112]. It implies the global system's capacity for linking actions across many different time scales of the constituent processes: there is no privileged time scale, and the system's temporal performance is self-similar at any scale. This property endows the system with the ability to respond adaptively to perturbations (external events) over a wide range of their temporal patterns, and enables adaptation to impinging neural impulse trains that vary unpredictably over a wide range of time scales [113] [114].
In the application cited in the foregoing, the systems were sufficiently small so that power laws with only one exponent were considered adequate. Hence, they fall into the category of monofractals. Granting, however, the plausibility of the suggested approach, we consider it necessary to introduce a refinement: characteristic time scales of the NMS component processes known to extend over a 1000-fold range (as stated earlier) render fitting a power law function with only one exponent unlikely. Accordingly, several power law functions with different exponents, each covering a section of the entire spectrum of scales, are required. This places NMS into the category of multifractals [115] [116], commonly thought to be indispensable for very large systems (e.g., geophysics [117]) but also successfully applied in numerous biological systems [112] [93]. This underscores the wide range of temporal scales to which systems with fractal characteristic can successfully adapt.
Patterns of natural stimuli and of neural impulse sequences intrinsic to the nervous system can vary unpredictably over a wide range of time scales, precluding prediction and anticipation of future events. The section “Linking actions across many scales” in [113] reviews numerous examples of neuronal structures that enable meeting this requirement, due to the fact that scale-invariant neural impulse trains adjust the dynamics of the recipient neuronal structures. This adaptation originates automatically from the basic properties of the neuronal processes being fractal and self-similar. By virtue of covering a wide spectrum of power law exponents, multifractals ensure sensitivity for unpredictable signals over a wider range of frequencies across multiple time scales [118] [119] and provide preparatory tuning to impending stimuli [120] [121]. Consequently, NMS provides considerable adaptive advantage for faithful processing of neural signals in unpredictable and uncertain environments.
Given that the function of NMS is to satisfy the requirements stipulated in the foregoing, it is, from an engineering point of view, surely extremely clumsily designed, with many redundancies and duplications of functions. Why is this so? It gives the impression that NMS in its present state evolved in stages, one stage superimposed on the other as if to attain a progressively higher degree of robustness and stability for ensuring secure contact with an ever-changing and unpredictable environment: perhaps many stages of consecutive “tinkering”; yet, seemingly preserving modular semiautonomy.
Based on an overview of neuroglia, extracellular fluid, and neuromodulation, which are commonly considered as separate topics, we propose characterizing their joint function in the framework of complex nonlinear dynamic system theory. We view the continuous temporal flux of this global system's activity as determining the characteristic parameters of neuronal function which, itself, participates in the system's dynamics. Our analysis of this temporal dynamics leads to the conclusion that it is multifractal, implying a high degree of sensitivity and adaptability to the temporal flux of constantly and unpredictably changing neural impulse traffic.