Abstract
In a previous study, we reported evidence for correlations between the firing of postsynaptic dorsal column (PSDC) neurons and cuneate neurons with overlapping receptive fields on the glabrous skin of the raccoon forepaw. The evidence was based on crosscorrelation and frequency response analyses of spontaneously firing neurons. However, crosscorrelation without white noise Gaussian analog inputs or Poisson distributed pulse train inputs is difficult to interpret because of the inherent convolution with the autocorrelation of the unknown input signals. While the data suggested positive correlations in the spinocuneate direction for most neuron pairs, we could not estimate the temporal characteristics of these putative connections. We have now reanalyzed these data using a parallelcascade method to estimate the first and secondorder kernels of a Volterra series approximation to the spinocuneate system. This unbiased analysis suggests that a positive correlation occurs after about 5 ms, probably followed by a negative correlation at about 12 ms. Secondorder kernels also had repeatable structure, indicating dual pathways with time separations of at least 10 ms.
INTRODUCTION
Postsynaptic dorsal column (PSDC) neurons are interneurons of the dorsal columns that receive mono and polysynaptic inputs from primary somatic afferents (AngautPetit 1975; Uddenberg 1968). Some PSDC neurons project to the cuneate nucleus (Cliffer and Willis 1994;Pierce et al. 1990) and could provide a secondary source of input to dorsal column nucleus projection neurons. However, attempts to establish anatomical or functional connections between PSDC and cuneate neurons are technically difficult and have not yet proved conclusive.
The raccoon provides an excellent model for approaching this problem because it has a large glabrous surface on the forepaw and high tactile acuity (Löhmer 1975). In an earlier study, we recorded extracellularly from pairs of PSDC and cuneate neurons in the raccoon under several stimulus conditions (Dick et al. 2001). Neuron pairs with overlapping receptive fields showed highly correlated firing when stimulated in the receptive field, as would be expected, but it is difficult to establish any functional connections other than the direct mutual input under these conditions. Recordings from such pairs during spontaneous activity also showed evidence of correlated firing, but we were unable to establish reliable timing of the correlations.
We have now reanalyzed the data obtained from spontaneously active pairs of PSDCcuneate neurons with overlapping receptive fields using a recently developed nonlinear approach. Parallel cascade identification was used to obtain the first and secondorder Volterra kernels of an unknown system comprising PSDC neurons as the input and cuneate neurons as the output. The firstorder kernels indicated weak but consistent linear excitation followed by inhibition during 5–15 ms after each spike in the PSDC neurons. Secondorder kernels also contributed to the system but did not simply reflect a nonlinear component of the firstorder signal. Instead, they suggested that information reaching or originating in PSDC neurons also travels to cuneate neurons by two pathways having a relative time separation of at least 10 ms.
METHODS
Recordings
Full details of the animal preparation, neuron identification and recording methods have been given previously (Dick et al. 2001). Adult raccoons (Procyon lotor) were initially anesthetized with ketamine (100 mg im) and halothane (2%) to allow insertion of a venous catheter. Surgery (craniotomy/laminectomy) and all recordings were done while the animal was anesthetized with αchloralose, 5 mg/ml, at approximately 1 ml/h. Extracellular recordings were made of PSDC and cuneate neuron pairs from 23 animals using ParyleneC insulated tungsten electrodes (1–2 MΩ impedance, AM Systems, Everett, WA). Animals were killed by an overdose of αchoralose plus pentobarbital, then perfused transcardially with saline (0.9%) followed by paraformaldehyde. All procedures were approved by the Dalhousie University ethics committee and were in accordance with the National Institutes of Health Guide for the Care and Use of Laboratory Animals, and the Canadian Committee on Animal Care.
Data analysis
The original recordings were obtained by computer controlled repeat sequences in which several stimulation protocols were used. These recording cycles included periods of 300 ms during which no stimulation was presented, to observe spontaneous activity. These records were processed to separate the 300ms spontaneous periods and concatenate them, together with silent periods of 150ms duration, to form a continuous record of spontaneous activity. The 150ms silent periods were added to prevent any spurious correlations between the originally separated recording periods.
Action potentials, recorded as 1 or 0 with a resolution of 0.1 ms, were convolved with a Sin(x)/x function (Fig. 2) using the FrenchHolden algorithm (French and Holden 1971;Peterka et al. 1978) and resampled with a resolution of 2 ms to prevent aliasing. The resulting records were analyzed as an unknown nonlinear dynamic system with the PSDC neuron signal as the input, x(t), as a function of time, t,and the cuneate neuron signal as output, y(t), represented by the first three terms of a Volterra series
Kernel estimation
Several methods have been developed for kernel estimation (French and Marmarelis 1999). Earlier methods relied on stimulating the unknown system with Gaussian white noise, or a close approximation, but more recent methods avoid this requirement. The parallelcascade method (Korenberg 1991) is based on the principle that a wide range of nonlinear dynamic systems can be approximated by a parallel cascade of simple nonlinear systems (branches), each consisting of a linear filter followed by a static nonlinearity (Fig. 1). Branches are added one by one, with each chosen to minimize the squared error remaining after subtracting the sum of all the previous branch outputs from the actual system output. Addition of branches proceeds until some predetermined criterion of error level or number of branches is reached.
The choice of linear filters is arbitrary. Filters chosen by crosscorrelation between input and output can produce very rapid convergence (French and Marmarelis 1999;Korenberg 1991). We have used an alternative approach, where the filters are formed from Gaussian distributed random numbers (French et al. 2001). This method is relatively slow and often requires up to 10^{5} cascade branches, but it is completely general and makes no initial assumptions about the forms of the kernels or the nature of the input or output signals. Modern computing power allows this inefficient, but general and objective method to be easily applied to a wide range of data.
Once each filter is chosen, the polynomial function is calculated by linear regression. Kernels are then updated by adding the filter function multiplied by the appropriate polynomial coefficients. An important feature of the method is that it can be applied to systems containing relatively highorder nonlinearities by extending the polynomial function. However, it is not necessary to construct all of the higherorder kernels. Another important point is that the final number of parameters used to fit the data are given by the kernel values themselves, so a large number of cascade branches does not increase the number of fitted parameters, only their accuracy.
After kernel estimation, percentage mean square error (MSE) values (French and Marmarelis 1999) were calculated for the zero and firstorder kernels alone and for the combined, zero, first, and secondorder kernels from
RESULTS
Data were analyzed from 26 pairs of PSDC and cuneate neurons with overlapping receptive fields on glabrous skin of the forepaw. The complete record lengths varied from 24 to 188 s with a mean of 82 s, including the additional 150ms gaps that were added to each 300ms recording period. PSDC neurons had a range of 0.3–21.3 action potentials per second (AP/s) and a mean rate of 5.9 AP/s. Cuneate neurons had a range of 1.2–75.8 AP/s and mean rate of 18.5 AP/s.
Parallelcascade fitting of first and secondorder kernels was performed for a period of 50 ms before and after the zero time lag between the signals until the residual square error between input and output (French and Marmarelis 1999) did not change in the third decimal place by addition of further cascades. This was typically achieved after about 75,000 cascades and never less than 60,000. At the completion of fitting, MSE values for the zero and firstorder kernels were in the range: 100.1–114.8% with mean: 103.5%. MSE for all kernels was: 88.4–99.8% with mean: 95.7%. It should be noted that estimating MSE causes difficulties with action potential output signals because the output appears in the denominator (Eq. 2 ) and the function can theoretically approach infinity when the output is zero. However, it is the most widely used measure of fitting Volterra and Wiener kernels, and it is clear that addition of the secondorder kernel produced a consistent reduction in MSE, indicating the presence of secondorder correlations between the two signals.
Kernel estimates were tested for their ability to predict the cuneate signal from the PSDC signal, using Eq. 1. As expected, firstorder kernels were unable to reliably predict the cuneate signal, although there was some indication of action potential timing in the predicted signal (Fig. 2) and addition of the secondorder kernel clearly contributed to the prediction.
Firstorder kernels
All kernels were normalized to a maximum range of ±1 for presentation and combination, but the zero level was not changed, so positive values represent excitatory connections and negative values inhibitory connections throughout. Firstorder kernels,K _{1}(τ), were noisy (Fig. 2), but there was usually evidence of a positive deflection at 5–10 ms. This could be seen more easily by stacking all of the normalized firstorder kernels together to make a perspective plot (Fig.3) in which a positive ridge can be seen at about 5 ms, followed by a negative valley at about 12 ms. A contour plot of all the firstorder kernels (Fig.4) shows that these two features are the only ones that are consistent through the entire set. Although numerous other peaks and depressions occurred, they were not consistently seen in all kernels, or even in the majority of kernels, but instead appeared to be randomly distributed at many different time lags, both before and after zero. The firstorder kernels may be considered to represent the probability of seeing action potentials in the cuneate neurons at times before and after an action potential in a PSDC neuron. Negative times were included to allow for the possibility of cuneate to PSDC interactions or signals from mutual thirdparty sources that arrive at cuneate neurons earlier than at PSDC neurons. However, there was no consistent evidence of excitatory or inhibitory interactions before time 0.
Secondorder kernels
Secondorder kernels were also noisy, and perspective plots were hard to interpret, but contour plots revealed a surprising and consistent pattern (Fig. 5). Peaks in the kernels occurred along diagonal tracks that were parallel to the main diagonal (τ_{1} = τ_{2}) and left a relatively flat region, or plain, around the main diagonal to a distance of about 10 ms on either side. Note that secondorder kernels of real systems are always symmetric around the main diagonal because the order of presentation of the two time lag variables is arbitrary. The edges of the diagonal plain were lined by bands of peaks and depressions that were not consistently located, except for their diagonal distributions, and that usually extended to both positive and negative time lags, up to the limits of the plots at ±50 ms. In some cases, there were several such parallel diagonal bands, and in others the central plain was wider, but the general features of a central flat plain lined by diagonal bands were seen consistently.
DISCUSSION
The data presented here do not show clear and unambiguous functional connections between PSDC neurons and cuneate neurons, but they do suggest that such connections exist and that they are more complex than previously envisaged. The analysis methods that we used were unbiased by any prior assumptions about the statistical properties of the action potential signals or the forms of any relationships between the two signals, yet they produced evidence that such relationships exist and gave approximate estimates for their timing. It is particularly interesting that the secondorder kernels were not simple products of the firstorder kernels. This is often the case for signals passing through compressive or facilitatory nonlinearities, where the secondorder term of a polynomial approximation to the nonlinear function occurs along the main diagonal (Marmarelis and Marmarelis 1978).
As described in methods, the use of Gaussian random numbers for the linear filters of the cascade branches required very large numbers of branches, in the range of 10^{4}10^{5}, but this did not change the number of parameters used to fit the data, which was given by the numbers of elements in the kernels. We consider the benefit of completely objective kernel estimation to be well worth the inefficiency of the method, and particularly important in the present situation, where the nature, or even existence of the putative connections was unknown.
The firstorder kernels indicate that activity in PSDC neurons is followed after about 5 ms by excitation and then inhibition of cuneate neurons, which decays to zero after about 15–20 ms. This is compatible with our previous finding that crosscorrelation measurements most often indicate an excitatory PSDC to cuneate flow of information (Dick et al. 2001) and agrees well with the minimum of 3 ms required for an action potential to travel between the two sites. The apparent inhibition could reflect a refractory period following cuneate firing or the action of some other local cuneate circuitry. An alternative explanation for the firstorder kernels could be an independent signal that excites both PSDC and cuneate neurons but with a relative delay of about 5 ms. However, this seems unlikely to be a descending source because the cuneate excitation occurs after the PSDC. The firstorder kernel data showed no consistent evidence for a cuneate to PSDC connection, so a simple PSDC to cuneate connection is clearly the most parsimonious explanation for the data.
As discussed previously (Dick et al. 2001), the PSDC to cuneate connection could result from PSDC neurons providing an excitatory input to cuneate neurons because some PSDC fibers terminate in the vicinity of cuneothalamic neurons (Cliffer and Willis 1994; Pierce et al. 1990) and most PSDC terminals in the rat cuneate nucleus have asymmetric synaptic specializations characteristic of excitatory synapses (de Biasi et al. 1995). The latency of about 5 ms observed here is consistent with this model.
The secondorder kernel results were more complex than expected. The major features to be explained are the flat central diagonal plain region, about 10 ms wide on either side of the diagonal τ_{1} = τ_{2}, and the parallel diagonal bands of peaks and depressions, which covered most of the measured time lag range of ±50 ms. The value at any point, τ_{1}, τ_{2}, in a secondorder kernel represents the average contribution to the output signal at time, t, that results from multiplying the input signal at time t  τ_{1} by the input signal at time t  τ_{2}. Because the input signal here is a train of action potentials in a PSDC neuron, we can consider the cuneate output that is produced by pairs of action potentials at different times. Therefore the flat central plain means that a pair of PSDC action potentials separated by about 0–10 ms do not make any major secondorder contributions to the cuneate signal, no matter what their timing relative to action potentials in the cuneate (at least within ±50 ms). To eliminate the possibility that pairs of PSDC action potentials do not actually occur with interspike intervals below 10 ms, we constructed interval histograms of PSDC neurons (data not shown). These gave an approximately exponential decay of intervals from a minimum of <1 ms, so the diagonal plain is not due to an artifact of the PSDC action potential timing.
Pairs of action potentials separated by about 10 ms or more did make significant secondorder contributions, and those contributions had a longlasting effect, but the exact timing, direction, and amplitude of the contributions cannot yet be defined. This secondorder contribution could reflect the activity of a third group of neurons that drive both PSDC and cuneate neurons, with appropriate conduction or synaptic delays, and the occurrence of diagonal bands at negative time lags makes this probable. Possible sources of this dual input include the somatosensory cortex, periaqueductal gray, the medial reticular formation, and the brain stem lateral reticular formation (discussed inDick et al. 2001). Of these, descending information from S_{1} cortex to dorsal column nuclei has been suggested previously by nonlinear systems analysis in the cat (Sclabassi et al. 1986).
The present analysis suggests that general nonlinear analysis can be useful for identifying complex neuronal connections under noisy conditions in the somatosensory system and elsewhere. However, much remains to be done. It will be important to gather longer records under standard conditions to improve kernel estimates and to develop statistical methods for reliably determining the probability that such connections exist, as well as their timing and effects.
Acknowledgments
Support for this work was provided by grants from the Canadian Institutes of Health Research to A. S. French and D. D. Rasmusson.
Footnotes

Address reprint requests to: A. S. French (Email:andrew.french{at}dal.ca).
 Copyright © 2002 The American Physiological Society