7.1. Aim
Another fundamental cortical property is selectivity for spatial frequency. Cortical cells inherit spatial frequency selectivity from geniculate cells (So and Shapley 1981), which acquire this selectivity from retinal ganglion cells. The ganglion cell‟s selectivity is due to the centre-surround organisation of its receptive field (Enroth- Cugell & Robson, 1966, Kuffler, 1953, Troyer et al., 1998). Cortical cells, however, are more sharply tuned to stimulus spatial frequency than are geniculate cells.
This chapter investigates the response of the model for this property and compares the model‟s performance with responses measured from cortical cells. In addition, the frequency histogram of the bandwidth of the model‟s tuning function is compared with the empirical results.
7.2. Spatial frequency tuning
Empirically, this property is examined by drifting an optimally oriented grating across the receptive field at a variety of spatial frequencies. The usual response measure is the fundamental Fourier amplitude of the impulse rate. The typical result (Movshon et al., 1978b), is that the neuron has an optimal spatial frequency, and that the response falls away rapidly on either side of the optimal value.
Neurons in the first cortical stage of the basic model produce a similar result, as illustrated in the Figure 7.1a.This Figure indicates the spatial frequency selectivity of the cell at the middle of the visual field patch. The peak of the curve is the response of the neuron for the optimal spatial frequency (0.49 cycle/deg) and the response falls monotonically on either side of the peak. The existence of an optimal value is easily understood. When the light bar of the grating is over the on-subfield and the dark bar is simultaneously over the off-subfield the two phases of the grating contribute constructively inmodulatingthecell‟simpulserate.
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The tightness of tuning to spatial frequency can be assessed from the bandwidth of the tuning curve at half height. A histogram of the spatial frequency tuning bandwidth for all active cells in the first cortical stage is plotted in Figure 7.1b (see the Methods chapter for the definition of active). Neurons in the basic model have bandwidths above 1.5 octaves, which is broader than their empirical counterparts (Movshon et al. 1978b). I speculated that the poor tuning is due to the lack of a surround mechanism in the subcortical stages. In particular, very low spatial frequencies produce substantial surround signals that antagonise centre signals (Enroth-Cugell and Robson, 1966). This should be noted that the majority of neurons in the Movshon study (Movshon et al. 1978b) were within 5 degree eccentricity and the model results are simulated at an eccentricity of 11 degree. This may account for some of the lack of agreement between model and empirical data.
Figure 7.1: Spatial frequency selectivity. (a) Spatial frequency tuning curve of the middle cell in the first cortical stage. A grating stimulus was drifted across the visual field patch and the fundamental frequency component of the response plotted against grating spatial frequency. The optimal spatial frequency for this cells is 0.49 cycle/deg and the bandwidth is 2.1 octaves. (b) Comparison of the bandwidth of the spatial frequency for active neurons in the first cortical stage of the basic model with simple cells (Movshon et al. 1978b).
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7.3. Improving the model
The histogram of the tuning bandwidth for the neurons in the first cortical stage indicated that the basic model is poorly tuned compared to physiological data. Many studies show that the spatial frequency tuning inherited from retinal ganglion cells is due to centre-surround organisation (Enroth-Cugell and Robson, 1966; Kuffler, 1953; Rodieck, 1965). The basic model includes no surround mechanism for ganglion and geniculate cells and the model was therefore modified by adding surround mechanism to the subcortical stages. With this modification, spatial frequency tuning became narrower (Figure 7.2a) due to suppression from the surround. Cortical cells are typically narrowly tuned with a bandwidth at half-height of 1.5 octaves (Movshon et al. 1978b) and the mean of the bandwidth for the cortical cells in the first stage is 1.3 octaves, close to the empirical data (Figure 7.2b).
Figure 7.2: (a) The green curve shows the spatial frequency tuning of the middle cell in the first cortical stage after adding a surround to the subcortical cells. The optimal spatial frequency is barely changed but the bandwidth of the selectivity is 1.1 octaves after adding surround mechanism. The red curve is the previous result from the basic model, for comparison. (b) The population histogram of bandwidth is in better agreement with empirical data (Movshon et al. 1978b).
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7.4. Discussion and conclusions
The model can reproduce the spatial frequency selectivity of cortical cells. The basic model showed a broadly tuned spatial frequency curve compared to physiological data. In order to rectify this problem, a surround mechanism was added to the basic model, making the cortical cells more sharply tuned for spatial frequency, similar to laboratory data. The model therefore fits with the idea that cortical spatial frequency selectivity comes from the convergence of the LGN inputs which they have centre- surround organisation. Some studies indicated that intracortical inhibition also plays a role in generating this selectivity (Zhu et al. 2010, Bauman and Bonds 1991). The static hyperpolarisation in the first cortical stage is the counterpart of this inhibition in this model. Most likely, intracortical inhibition is the source of this static hyperpolarisation as discussed in greater extent in section 8.3.
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