The Generation Of Electric And Magnetic Extracellular Fields

It is generally assumed that the neuronal events that cause the generation of electric and/or magnetic fields in a neural mass consist of ionic currents that have mainly postsynaptic sources. For these fields to be measurable at a distance from the sources, it is important that the underlying neuronal currents are well organized both in space and in time. The ionic currents in the brain obey Maxwell's and Ohm's laws.

The most important ionic current sources in the brain consist of changes in membrane conductances caused by intrinsic membrane processes and/or by synaptic actions. The net membrane current that results from changes in membrane conductances, either synaptic or intrinsic, can be either a positive or a negative ionic current directed to the inside of the neuron. These currents are compensated by currents flowing in the surrounding medium since there is no accumulation of electrical charge. Consider as the simplest case that of synaptic activity caused by excitatory postsynaptic currents (EPSCs) or inhibitory postsynaptic currents (IPSCs). Because the direction of the current is defined by the direction along which positive charges are transported, at the level of the synapse there is a net positive inward current in the case of an EPSC and a negative one in the case of an IPSC. Therefore, extracellularly an active current sink is caused by an EPSC and an active current source by an IPSC. Most neurons are elongated cells; thus, along the passive parts of the membrane (i.e., at a distance from the active synapses) a distributed passive source is created in the case of an EPSC and a distributed passive sink in the case of an IPSC. In this way, a dipole configuration is created (Fig. 1). At the macroscopic level, the activation of a set of neurons organized in parallel is capable of creating dipole layers. The following are important conditions that have to be satisfied for this to occur: (i) The neurons should be spatially organized with the dendrites aligned in parallel, forming palisades, and (ii) the synaptic activation of the neuronal population should occur in synchrony.

Lorente de No named the type of electric field created in this way an "open field,'' in contrast to the field generated by neurons with dendrites radially distributed around the soma which form a "closed field.'' In any case, as a result of synaptic activation, extracellular currents will flow. These may consist of longitudinal or transversal components, the former being those that flow parallel to the main axis of a neuron and the latter flow perpendicular to this axis. In the case of an open field, the longitudinal components will add, whereas the transversal components tend to cancel out. In the case of a closed field, all components will tend to cancel, such that the net result at a distance is zero.

The importance of the spatial organization of neuronal current sources for the generation of electric and/or magnetic fields measurable at a distance can be stated in a paradigmatic way for the cortex. Indeed, the pyramidal neurons of the cortex are lined up perpendicular to the cortical surface, forming layers of neurons in palissade. Their synaptic activation can occur within well-defined layers and in a synchronized way. The resulting electric fields may be quite large if the activity within a population of cells forms a

Figure 1 Model cortical pyramidal cell showing the patterns of current flow caused by two modes of synaptic activation at an excitatory (E) and an inhibitory (I) synapse. Typically, the apical dendrites of these cells are oriented toward the cortical surface. E, current flow caused by the activation of an excitatory synapse at the level of the apical dendrite. This causes a depolarization of the postsynaptic membrane (i.e., an EPSP), and the flow of a net positive current (i.e., EPSC). This current flow creates a current sink in the extracellular medium next to the synapse. The extracellularly recorded EPSP is shown on the left. It has a negative polarity at the level of the synapse. At the soma there exists a distributed passive current source resulting in an extracellular potential of positive polarity. I, current flow caused by activation of an inhibitory synapse at the level of the soma. This results in a hyperpolarization of the postsynaptic membrane and in the flow of a negative current. Thus, an active source is created extracellularly at the level of the soma and in passive sinks at the basal and apical dendrites. The extracellularly recorded IPSP at the level of the soma and of the apical dendrites is shown. Note that both cases show a dipolar source-sink configuration, with the same polarity, notwithstanding the fact that the postsynaptic potentials are of opposite polarity (EPSP vs IPSP). This illustrates the fact that not only does the nature of the synaptic potential determine the polarity of the potentials at the cortical surface but also the position of the synaptic sources within the cortex is important (adapted with permission from Niedermeyer and Lopes da Silva, 1999).

coherent domain (i.e., if the activity of the neuronal sources is phase locked). In general, the electric potential generated by a population of neurons represents a spatial and temporal average of the potentials generated by the single neurons within a macrocolumn.

A basic problem in electroencephalography/magne-toencephalography is how to estimate the neuronal sources corresponding to a certain distribution of electrical potentials or of magnetic fields recorded at the scalp. As noted previously, this is called the inverse problem of EEG/MEG. It is an ill-posed problem that has no unique solution. Therefore, one must assume specific models of the sources and of the volume conductor. The simplest source model is a current dipole. However, it should not be considered that such a model means that somewhere in the brain there exists a discrete dipolar source. It simply means that the best representation of the EEG/MEG scalp distribution is by way of an equivalent dipolar source. In the sense of a best statistical fit, the latter describes the centroid of the dipole layers that are active at a certain moment. The estimation of equivalent dipole models is only meaningful if the scalp field has a focal character and the number of possible active areas can be anticipated with reasonable accuracy. An increase in the number of dipoles can easily lead to complex and ambiguous interpretations. Nevertheless, methods have been developed to obtain estimates of multiple dipoles with only the a priori information that they must be located on the surface of the cortex. An algorithm that performs such an analysis is multiple signal classification (MUSIC), which is illustrated in Fig. 2 for the case of the cortical sources of the alpha rhythm. An alternative approach is to use linear estimation methods applying the minimum norm constraint to estimate the sources within a given surface or volume of the brain. Currently, new approaches are being explored that use combined fMRI and EEG/MEG recordings in order to create more specific spatial constraints to reduce the solution space for the estimation of the underlying neuronal sources. In general, the problems created by the complexity of the volume conductor, including the scalp, skull, layer of cerebrospinal fluid, and brain, are easier to solve in the case of MEG than EEG since these media have conductivities that affect the EEG much more than the MEG. The major advantage of MEG over EEG is the relative ease of source localization with the former. This means that when a dipole source algorithm is used on the basis of MEG recordings, a simple homogeneous sphere model of the volume conductor is usually sufficient to obtain a satisfactory solution. The position of the sources can be integrated in MRI scans of the brain using appropriate algorithms.

IV. DYNAMICS OF NEURONAL ELEMENTS: LOCAL CIRCUITS AND MECHANISMS OF OSCILLATIONS

Here, I discuss the dynamics of the electrical and magnetic fields of the brain, i.e., their time-dependent properties. These properties are determined to a large extent by the dynamical properties of ionic currents.

Figure 2 Results of a multiple signal classification analysis (MUSIC) scan of a 5-sec-long epoch of alpha activity recorded using a whole-head 151 MEG system during the eyes-closed condition. A singular value decomposition of the MEG signals yielded four factors. The scale indicates the cortical areas where the most significant sources were estimated (reproduced with permission from Parra et al, J. Clin. Neurophysiol. 17, 212-224, 2000).

Figure 2 Results of a multiple signal classification analysis (MUSIC) scan of a 5-sec-long epoch of alpha activity recorded using a whole-head 151 MEG system during the eyes-closed condition. A singular value decomposition of the MEG signals yielded four factors. The scale indicates the cortical areas where the most significant sources were estimated (reproduced with permission from Parra et al, J. Clin. Neurophysiol. 17, 212-224, 2000).

The most elementary phenomenon to be taken into account is the passive time constant of the neuronal membrane. This is typically about 5 msec for most cortical pyramidal neurons. Therefore, the activation of a synapse causing an ionic current to flow will generate a postsynaptic potential change that decays passively with such a time constant. However, there are other membrane phenomena that have much longer time constants. Some of these are intrinsic membrane processes, whereas others are postsynaptic. Among the intrinsic membrane phenomena, a multitude of ionic conductances, distributed along both soma and dendrites, enable neurons to display a variety of modes of activity. Among the synaptic phenomena, some present much longer time constants than the membrane passive time constant. This is the case, for example, for the GABAB-mediated inhibition that consists of a K + current with slow dynamics. The synaptic actions of amines and neuropeptides, in contrast with those of amino acids such as glutamate and GABA, also have slow dynamics. The effect of acetylcholine (ACh) is mixed since fast and slow actions occur, depending on the type of receptors to which ACh binds. Furthermore, note that the den-drites do not consist of simple passive membranes, as in the classical view; rather, the dendrites also have voltage-gated ion conductances that can contribute actively to the electrical behavior of the whole neuron. Thus, a neuron must be considered as a system of functional nodes, where each node represents a chemical synapse, an active ionic conductance, a metabolic modulated ion channel, or, in some cases, a gap junction (i.e., a direct electrical coupling between adjacent cells through a low-resistance pathway).

The sequence of hyperpolarizations and depolarizations caused by inhibitory and excitatory synaptic activity, respectively, can induce the activation or the removal of inactivation of intrinsic membrane currents. This aspect is difficult to analyze in detail under natural conditions in intact neuronal populations. However, much was learned about this kind of phenomena from studies carried out in vitro both in isolated cells and in brain slices. To illustrate this, consider the typical behavior of thalamic neurons that participate in the thalamocortical circuits with feedforward and feedback connections (Fig. 3). Even in isolation, these cells may show intrinsic oscillatory modes of activity, either in the alpha range of activity (namely, in the form of spindles at 7-14 Hz) or in the delta frequency range (0.5-4 Hz). To understand how

Human Brain Frequencies

Figure 3 Corticothalamic circuits and different types of neuronal activities. A thalamocortical relay cell is shown on the lower right-hand side, a neuron of the reticular nucleus (GABAergic) is shown on the left, and a cortical pyramidal cell is shown on top. The direction of the flow of action potentials along the axons is indicated by arrows. Positive signs indicate excitatory synapses and negative signs indicate inhibitory synapses. The insets show cellular responses obtained intracellularly. In this case, the activity of the cortical neuron was obtained by electrical stimulation of the intralaminar nucleus of the thalamus. At the membrane potential of—55 mV the response consisted of an antidromic spike (a) followed by a bursts of orthodromic spikes (o). At a more hyperpolarized level (—64 mV), the antidromic response failed but the orthodromic response consisted of a subthreshold EPSP. The responses of the two thalamic neurons were obtained by stimulation of the cortex. The response of the neuron of the reticular nucleus consists of a high-frequency burst of spikes followed by a series of spindle waves on a depolarizing envelope. The response of the thalamocortical neuron consists of a biphasic IPSP that leads to a low-threshold calcium spike (LTS) and a sequence of hyperpolarizing spindle waves (reproduced with permission from Steriade, 1999).

Figure 3 Corticothalamic circuits and different types of neuronal activities. A thalamocortical relay cell is shown on the lower right-hand side, a neuron of the reticular nucleus (GABAergic) is shown on the left, and a cortical pyramidal cell is shown on top. The direction of the flow of action potentials along the axons is indicated by arrows. Positive signs indicate excitatory synapses and negative signs indicate inhibitory synapses. The insets show cellular responses obtained intracellularly. In this case, the activity of the cortical neuron was obtained by electrical stimulation of the intralaminar nucleus of the thalamus. At the membrane potential of—55 mV the response consisted of an antidromic spike (a) followed by a bursts of orthodromic spikes (o). At a more hyperpolarized level (—64 mV), the antidromic response failed but the orthodromic response consisted of a subthreshold EPSP. The responses of the two thalamic neurons were obtained by stimulation of the cortex. The response of the neuron of the reticular nucleus consists of a high-frequency burst of spikes followed by a series of spindle waves on a depolarizing envelope. The response of the thalamocortical neuron consists of a biphasic IPSP that leads to a low-threshold calcium spike (LTS) and a sequence of hyperpolarizing spindle waves (reproduced with permission from Steriade, 1999).

oscillations can take place in these circuits it is important to realize that these cells have, among other kinds of ionic currents, a low-threshold Ca2+ conductance (1T) that contributes to the low-threshold spike. This 1T conductance is de-inactivated by a previous membrane hyperpolarization and causes sufficient depolarization of the cell for the activation of a persistent (non-inactivating) Na+ conductance. These cells also have a nonspecific cation "sag" current (Ih) that has much slower kinetics than 1T and is activated by hyperpolarization. The alpha range oscillatory mode depends essentially on the low-threshold Ca2+ current 1T. In addition, other currents contribute to the oscillatory behavior—namely, a fast transient potassium current (1A) that has a voltage dependence that is similar to that of 1T, a slowly inactivating potassium current, and calcium-dependent potassium currents that need increased intracel-lular Ca2+ concentration for activation—and are mainly responsible for after-hyperpolarizations. In this way, a sequence of hyperpolarizations followed by depolarizations tends to develop. This could be sufficient for such cells to behave as neuronal "pacemakers." Under the in vitro conditions in which these basic properties have been studied, the initial hyper-polarization that is necessary for the removal of the inactivation of the Ca2+ current is provided artificially by an intracellular injection of current. However, under natural conditions in the intact brain, the oscillatory behavior cannot occur spontaneously (i.e., it is not autonomous). For the de-inactivation of the low-threshold Ca2+ current to occur, initially the cell has to be hyperpolarized by a synaptic action mediated by GABAergic synapses that impinge on these cells and stem from the GABAergic neurons of the reticular nucleus of the thalamus. Therefore, the notion that these thalamo-cortical relay cells behave as pacemakers, in the sense of generating pure autonomous oscillations, in vivo is only a relative one. Indeed, they need a specific input from outside to set the conditions under which they may oscillate. Thus, in vivo the alpha spindle and delta waves result from network interactions. Nevertheless, the intrinsic membrane properties are of importance in setting the frequency response of the cells and of the network to which they belong.

In general terms, it must be considered that neurons, as integrative units, interact with other neurons through local circuits, excitatory as well as inhibitory. In a neuronal circuit feedforward and feedback elements have to be distinguished. In particular, the existence of feedback loops can shape the dynamics of a neuronal network affecting its frequency response and even creating the conditions for the occurrence of resonance phenomena and other forms of oscillatory behavior. Furthermore, note that the transfer of signals in a neuronal network involves time delays and essential nonlinearities. This may lead to the appearance of nonlinear oscillations and possibly even to what is in mathematical terms called chaotic behavior.

In recent years, much has been learned about these issues with the advent of the theory of nonlinear dynamics. This has resulted in the application of nonlinear time series analysis to EEG signals, with the aim of estimating several nonlinear measures to characterize different kinds of EEG signals. This matter is the object of recent interesting studies reported in specialized publications in which the question ''chaos in brain?" is discussed. I stress here only that a theoretical framework based on the mathematical notions of complex nonlinear dynamics can be most useful to understanding the dynamics of EEG phenomena. In this context, insight may be obtained into how different types of oscillations may be generated within the same neuronal population and how such a system may switch from one type of oscillatory behavior to another. This occurs, for example, when an EEG/MEG characterized by alpha rhythmic activity suddenly changes into a 3-Hz spike and wave pattern during an absence seizure in epileptic patients. Based on the use of mathematical nonlinear models of neuronal networks, it is possible to formulate hypotheses concerning the mechanisms by means of which a given neuronal network can switch between these qualitatively different types of oscillations. This switching behavior depends on input conditions and on modulating parameters. Accordingly, such a switch can take place depending on subtle changes in one or more parameters. In the theory of complex nonlinear dynamics, this is called a bifurcation. In this respect, the most sensitive parameter is the neuronal membrane potential that in the intact brain is modulated by various synaptic inputs. Typically, the change of oscillation mode may be spectacular, whereas the initial change of a parameter may be minimal.

The frequency of a brain oscillation depends both on the intrinsic membrane properties of the neuronal elements and on the properties of the networks to which they belong. Considerations regarding the membrane conditions that determine which ionic currents can be active at a given time lead to the conclusion that the modulating systems mediated by several neurochemical systems are of utmost importance in setting the initial conditions that determine the activity mode of the network (i.e., whether it will display oscillations or not and at which frequency). Different behavioral states are characterized by the interplay of different chemical neurotransmitter and neuromodulator systems.

A fundamental property of a neuronal network is the capacity of the neurons to work in synchrony. This depends essentially on the way the inputs are organized and on the network interconnectivity. Thus, groups of neurons may work synchronously as a population due to mutual interactions. The experimental and theoretical work of Ad Aetsen and Moshe Abeles showed the existence of precise (within 5 msec) synchrony of individual action potentials among selected groups of neurons in the cortex. These synchronous patterns of activity were associated with the planning and execution of voluntary movements. These researchers noted that during cognitive processes the neurons tend to synchronize their firing without changing the firing rates, whereas in response to external events they synchronize firing rates and modulate the frequency at the same time. In addition, they also showed that the synchronization dynamics are strongly influenced by the level of background activity. This indicates the importance that the ongoing electrical activity can have in setting the activity climate in a given brain system.

A fundamental feature of the cortex is that groups of neurons tend to form local circuits organized in modules with the geometry of cortical columns. The basic cortical spatial module is a vertical cylinder with a 200-300 mm cross-section. There exist different systems of connecting fibers between cortical columns, namely, (i) the collaterals of axons of pyramidal neurons that may spread over distances of approximately 3 mm and are mostly excitatory; (ii) the ramifications of incoming terminal axons that may extend over distances of 6-8 mm along the cortical surface; and (iii) the collaterals of interneurons, an important part of which are inhibitory, that may branch horizontally over 0.5-1 mm within the cortex. These systems range over distances on the order of magnitude of hundreds of micrometers, and this determines the characteristic length of intracortical interactions.

It is not simple to directly relate the dynamic behavior of a neuronal network to the basic parameters of neurons and synapses. In order to construct such relationships, basic physiological and histological data have to be combined like pieces of a puzzle. However, the available knowledge of most neuronal networks in terms of detailed physiological and even histological information is still incomplete. To supplement this lack of specific knowledge, a synthetic approach can be useful. This implies the construction ofconnectionist models ofneuronal networks using all available and relevant data. Such models can be of practical use to obtain a better understanding of the main properties of a network. Indeed, a model offers the possibility of studying the influence of different parameters on the dynamic behavior of the network and of making predictions of unexplored properties of the system, which may lead to new experiments. In this way, EEG signals, particularly local field potentials (LFPs), can also be modeled. Thus, hypotheses concerning the relevance of given neuronal properties to the generation of special EEG features may be tested.

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