A wavelet-based neural model to optimize and read out a temporal population code

Abstract

It has been proposed that the dense excitatory local connectivity of the neo-cortex plays a specific role in the transformation of spatial stimulus information into a temporal representation or a temporal population code (TPC). TPC provides for a rapid, robust, and high-capacity encoding of salient stimulus features with respect to position, rotation, and distortion. The TPC hypothesis gives a functional interpretation to a core feature of the cortical anatomy: its dense local and sparse long-range connectivity. Thus far, the question of how the TPC encoding can be decoded in downstream areas has not been addressed. Here, we present a neural circuit that decodes the spectral properties of the TPC using a biologically plausible implementation of a Haar transform. We perform a systematic investigation of our model in a recognition task using a standardized stimulus set. We consider alternative implementations using either regular spiking or bursting neurons and a range of spectral bands. Our results show that our wavelet readout circuit provides for the robust decoding of the TPC and further compresses the code without loosing speed or quality of decoding. We show that in the TPC signal the relevant stimulus information is present in the frequencies around 100 Hz. Our results show that the TPC is constructed around a small number of coding components that can be well decoded by wavelet coefficients in a neuronal implementation. The solution to the TPC decoding problem proposed here suggests that cortical processing streams might well consist of sequential operations where spatio-temporal transformations at lower levels forming a compact stimulus encoding using TPC that are subsequently decoded back to a spatial representation using wavelet transforms. In addition, the results presented here show that different properties of the stimulus might be transmitted to further processing stages using different frequency components that are captured by appropriately tuned wavelet-based decoders.

Keywords: Haar wavelets; pattern recognition; spike neural network; temporal coding; visual system; wavelet transform.

Figure 1

The TPC encoding model. In a first step, the input image is projected to the LGN stage where its edges are enhanced. In the next stage, the LGN output passes through a set of Gabor filters that resemble the orientation selectivity characteristics found in the receptive fields of V1 neurons. Here we show the output response of one Gabor filter as input for the V1 spiking model. After the image onset, the sum of the V1 network's spiking activity over time gives rise to a temporal representation of the input image. This temporal signature of the spatial input is the, so called temporal population code, or TPC.

Figure 2

The TPC encoding paradigm. The stimulus, here represented by a star, is projected topographically onto a map of interconnected cortical neurons. When a neuron spikes, its action potential is distributed over a neighborhood of a given radius. The lateral transmission delay of these connections is 1 ms/unit. Because of these lateral intra-cortical interactions, the stimulus becomes encoded in the network's activity trace. The TPC representation is defined by the spatial average of the population activity over a certain time window. The invariances that the TPC encoding renders are defined by the local excitatory connections.

Figure 3

Computational properties of the two types of neurons used in the simulations: regular (RS) and burst spiking (BS). The RS neuron shows a mean inter spike interval of about 25 ms (40 Hz). The BS type displays a similar inter-burst interval with a within burst inter-spike interval of approximately 7 ms (140 Hz) every 35 ms (28 Hz).

Figure 4

(A) Neuronal readout circuit based on wavelet decomposition. The buffer cells B1 and B2 integrate, in time, the network activity performing a low-pass approximation of the signal over two adjacent time windows given by the asynchronous inhibition received from cell A. The differentiation performed by the excitatory and inhibitory connections to W gives rise to a band-pass filtering process analogous to the wavelet detail levels. (B) An example of band-pass filtering performed by the wavelet circuit where only the frequency range corresponding to the resolution level Dc₃ is kept in the spectrum.

Figure 5

The stimulus classes used in the experiments after the edge enhancement of the LGN stage.

Figure 6

The stimulus set. (A) Image-based prototypes (no jitter in the vertices applied) and the globally most different exemplars with normalized distance equal one. The distortions can be very severe as in the case of class number one. (B) Histogram of the normalized Euclidean distances between the class exemplars and the class prototypes in the spatial domain.

Figure 7

Baseline classification ratio using Euclidean distance among the images from the stimulus set in the spatial domain.

Figure 8

Comparison among the correct classification ratio for different resonance frequencies of the wavelet filters for both types of neurons RS and BS. The frequency bands of the TPC signal is represented by the wavelet coefficients Dc₁ to Ac₅ in a multi-resolution scheme. The network time window is 128 ms.

Figure 9

Speed of encoding. Number of bits encoded by the network's activity trace as a function of time. The RS-TPC and BS-TPC curves represent the bits encoded by the network's activity trace without the wavelet circuit. The RS-Wav and BS-Wav correspond to the bits encoded by the wavelet coefficients using the Dc₃ resolution level for RS neurons and the Dc₅ for BS neurons, respectively. For a time window of 128 ms the Dc₃ level has 16 coefficients and the Dc₅ has only 4 coefficients. The dots in the figure represent the moment in time where the coefficients are generated.

Figure 10

Single-sided amplitude spectrum of the wavelet prototype for each stimulus class used in the simulations. The signals x(t) where reconstructed in time using the wavelet coefficients from the Dc₃ and Dc₅ levels for RS and BS neurons, respectively. The shaded areas shows the optimal frequency response of the Dc₃ level (62–125 Hz) and of the Dc₅ level (15.5–31 Hz). The less pronounced responses around 400 Hz are aliasing effects due to the signal reconstruction to calculate the Fourier transform (see discussion).

Figure 11

Prototype-based classification hit matrices. For each class in the training we average the wavelet coefficients to form class prototypes. In the classification process, the euclidean distance between the classification set and the prototypes are calculated. A stimulus is assigned to a the class with smaller euclidean distance to the respective class prototype.

Figure 12

Distribution of errors in the wavelet-based prototype classification with relation to the Euclidean distances within the prototyped classes.