## Abstract

Examples of multiplication by neurons or neural circuits are scarce, although many computational models use this basic operation. The owl's auditory system computes interaural time (ITD) and level (ILD) differences to create a two-dimensional map of auditory space. Space-specific neurons are selective for combinations of ITD and ILD, which define, respectively, the horizontal and vertical dimensions of their receptive fields. A multiplication of separate postsynaptic potentials tuned to ITD and ILD, rather than an addition, can account for the subthreshold responses of these neurons to ITD-ILD pairs. Other nonlinear processes improve the spatial tuning of the spike output and reduce the fit to the multiplicative model.

Space-specific neurons in the owl's auditory map of space represent the results of all computations involved in the determination of the ITD and ILD that define the owl's auditory space (1, 2). The parallel pathways that process these cues merge in the external nucleus of the inferior colliculus (ICx) to confer on the space-specific neurons selectivity for combinations of ITD and ILD (3, 4) (Fig. 1A). How do postsynaptic potentials (psps) from the two sources interact to produce subthreshold and suprathreshold responses to ITD-ILD pairs?

Clusters of similar ITD-ILD pairs elicited suprathreshold depolarizing psps, whereas other ITD-ILD pairs induced hyperpolarization (Fig. 1C) (5–8). These response properties underlie the center excitatory and surround inhibitory organization of receptive fields (Fig. 1D) (9). All the neurons studied behaved like analog AND gates of ITD and ILD, suggesting that the two inputs are multiplied instead of being added. We used the method of singular value decomposition (svd) to show multiplication (10). This technique transforms a matrix into a weighted sum of the products of two independent vectors, called “singular vectors.” The contribution of these products to the original matrix is ranked by the elements (“singular values”) of a diagonal matrix.

The following example illustrates how the method works using the data obtained from a space-specific neuron. The responses of the neuron to ITD-ILD pairs were arranged in a matrix, *M*, in which ILD and ITD varied along rows and columns, respectively (Fig. 2A). We tested the hypothesis that the matrix contains constant and multiplicative components as in the following equation:(1)where *V*
_{o} represents a membrane potential offset. The second term indicates the multiplicative interaction between ITD and ILD inputs. *U*(*i*) and*V*(*j*) are the ITD and ILD inputs and λ is a singular value. The svd of *M* yielded nearly constant first singular vectors, confirming that the dominant term in*M* was an offset membrane voltage*V*
_{o} (blue area in Fig. 2B). We determined the optimal *V*
_{o} value in the equation (*V*
_{o} = −66.8 mV in this example) by minimizing the sum of the squared singular values other than the first one in the svd of *M* − *V*
_{o}. This procedure maximizes the difference in fractional energy or weight between the first and the remaining singular values of*M* − *V*
_{o} and is equivalent to a least squares fit to the multiplicative model. For all neurons, the best *V*
_{o} was usually close to the maximal level of hyperpolarization, corresponding to the deepest trough of ITD curves. An svd of the sample matrix after*V*
_{o} subtraction showed 98.57% of the energy in the first singular value and 0.96% in the second. Thus, a major part of the data matrix consists of a constant offset membrane voltage *V*
_{o} and a multiplication product. Computation of *U*
_{1} and *V*
_{1}(first left and right singular vectors) of *M* −*V*
_{o} (Fig. 2, E and F) yielded functions similar to the neuron's measured ITD and ILD response curves, once*V*
_{o} was added (11) (Fig. 2, C and D). Finally, we verified that the equation provided a close approximation of the original matrix (Fig. 2B). This finding confirms that ITD and ILD inputs are independent of each other, as previously shown by other methods (3,4).

We tested the multiplicative hypothesis in 15 neurons in which we obtained responses for a large number of ITD-ILD pairs. All neurons showed similar trends to those described in the example mentioned above. After subtraction of *V*
_{o}, the first singular value yielded a mean fractional energy of 96.6 ± 2.1% (Fig. 3A). We also examined which lower ranking singular values (second, third, etc.) were above the variability of the data and within the confidence interval of the first singular value. For a confidence level of 99.9% (12) with 4 degrees of freedom and the largest standard deviation, all singular values other than the first one were within the noise of our measurement in seven neurons. In these cases, 100% of the energy was in the first singular value. The first and second singular values were above the noise level in the remaining eight neurons (Fig. 3B). In these cases, the energy in the first singular value was 97.9 ± 0.9%. Thus, multiplication accounts for a large part of the observed responses in all neurons.

We also tested an additive model for comparison with the multiplicative model. A pure additive combination of ITD and ILD inputs can be given by(2)
*V*
_{m} is a constant and corresponds to the mean membrane potential. *F*(*i*) and *G*(*j*) represent the ITD and ILD inputs and were obtained by averaging rows and columns of *M*– *V*
_{m}, respectively. However, this model could not reconstruct the original data matrix as well as the multiplicative model (Fig. 3, C and D). The error of the fit to the original data was significantly larger for the additive model than for the multiplicative one (*F* test, *P* < 0.002). We also considered the possibility of both multiplication and addition contributing to the subthreshold responses of space-specific neurons. However, a combination of addition and multiplication turned out to be nothing but multiplication. Therefore, the above comparison of the multiplicative and additive models should suffice to show which model explains the data better.

For comparison with the intracellular data, we performed an svd on the spike data for the same 15 cells that were used for the analysis of psps (Fig. 4A). One of these neurons fired a single spike in response to the 100-ms stimulus and behaved like a perfect digital AND gate in that 100% of the energy was in the first singular value. However, lower ranking singular values varied from neuron to neuron more widely in the spike data than in the membrane potential data. This variability caused the mean fractional energy for the first singular value to drop to 78.6 ± 40.7% (*n* = 15 neurons). Thus, the distribution of energy was more clearly biased toward the first singular value in the intracellular data than in the spike data. This departure from multiplication is partly due to additional nonlinear processes such as thresholding and nonlinear conversion of membrane potential to spikes (7) (Fig. 5). We also compared the multiplicative and additive models on the spike data. The reconstructed matrix fitted the data matrix better with multiplication than with addition (Fig. 4, B and C). The error of the fit was larger for the additive model than for the multiplicative one (*F* test,*P* < 0.002).

Examples of multiplication by neurons or neural circuits are scarce, although many computational models use this basic operation (13–16). Neurons of the monkey's posterior parietal lobe show “gain fields” that can be explained by a multiplication of retinal and eye or head position signals (17). The response of high-order visual neurons of locusts and pigeons predicts time of collision by multiplying the velocity of the retinal image edge with an exponential function of the size of the retinal image of the object (18, 19). Behavior and high-order visual neurons in flies show responses that are consistent with coincidence detection, although the primary coincidence detectors remain to be identified (20). None of these studies identified the relevant psps that were multiplied, with one exception, in which postsynaptic inhibition was shown to underlie the directional sensitivity of the rabbit's visual ganglion cells (21). Mel and Koch (22) put forth a model in which multiplication of lower order receptive fields gives rise to a higher order receptive field. The owl's space-specific neurons behave like their model, although we do not know what biophysical mechanisms are involved. γ-Aminobutyric acid–mediated inhibition and *N*-methyl-d-aspartate–mediated responses in ICx might contribute to the multiplicative process (23, 24).

↵* To whom correspondence should be addressed. E-mail jose{at}etho.caltech.edu