In biological tissues, there are no free electrons. Electric currents are due to ions, e.g., K+, Na+, Cl-, Ca2+, etc. These ions flow in response to the local electric field, according to Ohm's law, but also in response to their local concentration gradient, according to Fick's law (Plonsey 1969). In the resting state of the membrane, the concentration gradients and electric field are due to ion channel pumps, which use energy acquired from ATP to move ions across the membrane against their diffusion gradient.

The concentration of each ion inside and outside the membrane remains essentially constant in time. The transmembrane voltage, however, changes radically in time, the strongest example being the action potential. Thus for the purposes of discussing small scale neural activity, we take the transmembrane potential Vm as the primary dynamic state variable to be considered. By convention, Vm is defined as the potential inside relative to that outside, i.e., Vm = V; — Vo. The current per unit length im flowing across the membrane, rather than the transmembrane potential, is considered the basic source variable of the extracellular fields detected by EEG and MEG.

Currents flow in neurons when a neurotransmitter binds to receptors on ion channels located in the membrane. Detailed consideration of a three-dimensional dendrite or axon has shown that the phenomenon of current flow in the cell may be well described by a one-dimensional approximation. The same mathematical treatment applies to undersea telegraph cables, comprised of an insulated metal core immersed in conductive salt water, thus the treatment is called "cable theory."

Assuming a membrane with conductive and capacitive properties, surrounded by fluids with conductive properties only, and applying current conservation in each compartment leads to

Tm ^dm = Am ~dXmr — (Vm — Er) — fm iother(x,t) (2.1)

where the membrane time constant Tm = rmcm, the neuron space constant Am = y/rm/(r; + ro), the membrane resistance times unit area rm = 1/gm, and Er is the resting transmembrane potential. Mathematically this is identical in form to the heat equation, which governs the one-dimensional flow of heat in a heat-conducting rod. It has been studied extensively and has well-known analytic solutions for appropriate boundary conditions.

Consider an isolated dendrite of length L, with current injected at one end. A solution may be derived for Vm(x). The corresponding transmembrane current per unit length im(x) may be written im (x) ~—Io S(x) + < /^/inj e-x/Xm, x > 0 (2.2)

Fig. 2. (a) Cylindrical cable representing a segment of dendrite or axon. A single compartment is shown with transmembrane current per unit length im. (b) Transmembrane current per unit length im at steady-state, due to current injected at x = 0. (The delta function at x = 0 is not drawn to scale; the actual area of the delta function equals the area of the positive (outward) current flow.)

where 5(x) is the Dirac delta function. Figure 2(b) shows the solution for x > 0. The delta function in Fig. 2(b) is not drawn to scale; the integral of (2.2) over all x is zero. Section 4 shows that, far away from the cable, the potential $ may be approximated as if it were generated by an ideal dipole, consisting of a point source and sink separated by a distance Am. This transmembrane current, driven not so much by the transmembrane potential as by difffusion, implies a nonzero extracellular electric field through current conservation.

Neuronal currents of this sort generate extracellular electric and magnetic fields, which are detected using EEG and MEG. The fields generated by a single neuron are much too small to be detected at the scalp, but the fields generated by synchronously active neurons, with advantageous geometric alignment, can be detected. Stellate cells have approximately spherical dendritic trees, so far away the extracellular fields tend to add with all possible orien-tiations and cancel. Pyramidal cells have similar dendritic trees, but the tree branches are connected to the cell body (soma) by a long trunk, called the apical dendrite. It is a fortuitous anatomical feature of the cortex that pyramidal cells have their apical dendrites aligned systematically, along the local normal to the cortical surface. In this way, the fields of pyramidal neurons superimpose geometrically to be measurable at the scalp.

Fig. 2. (a) Cylindrical cable representing a segment of dendrite or axon. A single compartment is shown with transmembrane current per unit length im. (b) Transmembrane current per unit length im at steady-state, due to current injected at x = 0. (The delta function at x = 0 is not drawn to scale; the actual area of the delta function equals the area of the positive (outward) current flow.)

Several factors contribute to the net fields measured at the scalp. Many active neurons and fortuitous alignment are not enough. As neuronal oscillations tend to oscillate with predominant frequencies, dependent upon functional activity, only synchronously active neurons will sum coherently in time (Elul 1972; Nunez 1981). Consider a 1 cm2 region of cortex, containing approximately 107 aligned pyramidal cells. Make the idealized assumption that all these neurons are oscillating at the predominant frequency, (e.g., 10 Hz resting rhythm). If only 1% of these neurons are synchronously active, i.e., oscillate in phase with each other, and 99% are oscillating with random phase. If the contribution from the asynchronous neurons may be treated as Gaussian then, because N unit-variance Gaussian random numbers sum to Vn , the relative contribution of synchronous to asynchronous neurons would be 105/%/107 ~ 30. Thus scalp EEG and MEG are considered to be dominated by synchronous neural activity. Indeed, amplitude reduction in EEG clinical and research circles is often termed desynchronization. Of course, phase desyn-chronization is only one of many possible mechanisms that could reduce the amplitude of the net voltage at the scalp. Alternatively, synchronous spike input to a patch of cortex can generate event-related synchronization in the dendritic fields. Such phase synchronization is one mechanism for producing event-related potentials (Makeig et al. 2002).

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