We define the local field potential ( LFP) to be the total spiking activity of all units within a circle of radius r around any particular unit. In general, we use r=10, an area large enough to include inhibitory interactions (which occur on a ring 8 and 9 lattice constants away). Our definition of the LFP is a crude approximation of the signal recorded by low-pass filtering the electrical signal from a low-impedance micro-electrode, and does not include any dendritic or synaptic component. We will contrast this with the spiking activity from a single unit (single-unit activity or SUA).
Figure 6: The top graph shows the local field potential (LFP), computed by summing the spiking activity of all leaky integrate-and-fire units within a disk of radius r=9, as a function of time during a typical simulation. We superimposed (dashed line) the spikes from a representative unit in the same part of the network. The average single unit activity (SUA) power spectrum (middle graph) shows such peak and is typical of the spectrum of a Poisson process with refractory period. The power spectrum of the LFP, shown in the bottom panel, shows a clear peak around 30 Hz. Although not shown, the variability in the interspike interval distribution is high, in accordance with experimental data.
Fig. 6a shows the LFP during a typical simulation run. The strongly fluctuating signal has a broad peak (25-45 Hz) in the power spectrum (Fig. 6c). This implies that the cells that contribute to the LFP are partially synchronized (at most only about 25 out of the 400 cells fire together), leading to broad oscillatory characteristics. The SUA tends to be aligned with the maxima of the LFP, as reported experimentally, without showing any significant evidence of periodic firing or a peak in the associated single-cell power spectrum (Fig. 6b).
To test whether this behavior is a trivial outcome for any model of spiking cells with feedback connectivity, we performed simulations using the same neural model, except for one change: each cell had excitatory and inhibitory synapses onto units chosen at random (independent of distance) on the lattice. The power spectra of the SUA and LFP are displayed in Fig. 7. The single cell power spectrum now has a large peak, implying that the discharge pattern of single cells is periodic. Such power spectra are in general not found in spike trains of non-bursting cells . Furthermore, the interspike interval variability under this condition is much lower than observed experimentally  in cells firing at medium or large rates. Because of the lack of spatial structure the system is ergodic; that is the population average (the LFP in Fig. 7) has essentially the same structure as the temporal average over a single unit (the SUA in Fig. 7).
Figure 7: The power spectrum of individual cells (upper panel) versus the spectrum of the LFP (lower panel) in a network with non-local random connectivity with both excitation and inhibition (everything else is identical to the previous figure). Here, both signals display a strong oscillatory component. The interspike interval variability is much lower under these conditions than observed empirically.
The oscillations in the LFP depend most strongly on the amount of lateral inhibition and the rate of external input in the model. Increasing leads to a much sharper spectral peak (especially for ) and to a small increase in the location of the peak. Increasing the input rate has a more pronounced effect, leading simultaneously to broadening of the peak and to an increase in the mean frequency.
The large, broadly periodic fluctuations in the LFP reflect partial synchronized and correlated discharge patterns of cells in a local population. We can explain the properties of the local field signal by reducing it to correlations of its constituents. Since we define the LFP to be the sum of all N single-unit spiking activity within a circle of radius r, the auto-correlation function of the LFP will be the sum of N auto-correlations functions of the individual cells and cross-correlation functions among individual cells within the circle,
In general, the auto-correlation of the LFP will be dominated by the cross-correlation functions among individual units and the power spectrum of the LFP---identical to the Fourier transform of the correlation function (Wiener-Khintchine theorem)---will in turn be dominated by the sum of the Fourier transforms of the individual cross-correlation functions. In order to characterize the power spectrum of the LFP, we will now describe the characteristics of the cross-correlations and their Fourier transforms.
Figure 8: Typical cross-correlation functions between neighboring pairs of mutually excitatory cells (top graph) and mutually inhibitory pairs separated by 9 lattice units. The solid lines indicate the best least square fit of eq. 3 to the data, with the fast initial decay (the ``tower") in fitted by an exponential with and the slower decay in both and fitted by another exponential with .
For mutually excitatory pairs of cells, the cross-correlation terms display a structure called a ``castle on a hill" in the neurophysiological literature , i.e., a 10 to 20 msec wide peak centered at zero (Fig. 8) followed by a slower decay to the asymptotic level of chance coincidences for long time lags. The peak of is always around zero and is generated by recurrent excitation. At higher values of inhibition , a small but significant secondary peak in the cross- correlation appears between the ``castle" and the slower decline. For cell pairs that are 8 or 9 units distant, the interaction will be (on average) mutually inhibitory and the associated correlation function is characterized by a gentle trough that rises slowly to the background level of chance coincidences at longer times. As discussed in Usher et al. (1994), the process governing the slow decay in the excitatory and the slow rise in the inhibitory correlation functions is the same.
The principal factor leading to a power spectrum peak in the 30-70 Hz range is the relative width of the excitatory cross-correlation peaks to the inhibitory troughs. In general, the excitatory "castles" are sharp relative to the broad dip in the cross-correlation due to inhibition. In Fourier space, these relationships are reversed: broader Fourier transforms of excitatory cross-correlations are paired with narrower Fourier transforms of inhibitory cross-correlations. Superposition of such transforms leads to a peak in the 30-50 Hz range. A less important factor is the existence (at higher values of inhibition) of a secondary peak around 20-30 msec in the excitatory cross- correlations.
The principle behind the formation of the power spectrum peak can be captured in the following simplified but generic example, which assumes that the excitatory and inhibitory correlation functions qualitatively behave as:
where the baseline of coincidence is normalized to 1. In Fig. 8, we fitted these equations against two particular cross-correlation functions from our network using and .
The power spectrum of the LFP, , will be dominated by the Fourier transforms of many such correlation functions, with
for . The terms a and b account for the strength and the number of excitatory and inhibitory cell pairs within the circle (with ).
Figure 9: Schematic of how the oscillatory peak in the LFP can be caused by a sharply tuned excitatory cross-correlation function superimposed onto a much broader inhibitory cross-correlation function . If the two are subtracted, the low frequency components cancel, leaving a peak at a non-zero frequency.
Consider now the simple case of , which will occur in our scenario if the LFP is taken over a circle with r > 9. In this case the spectrum increases from low frequencies, reaches a maximum at , and decays back to zero for large frequencies. The superposition of two representative cross-correlation transforms yielding the desired peak is shown in Fig. 9).