In this chapter, we address the role of horizontal connections in a layer of spiking cells in the processing of sensory information. Our focus is on the primary visual cortex, although some of the discussion is more general. The intrinsic irregularity of spike trains and the oscillatory nature of neural activity as measured by local field potentials provide us with constraints on the nature of lateral recurrent interactions in cortical layers. Once we have shown how these constraints can be satisfied, we will explore possible neural computations performed by the lateral circuits within the visual cortex. Some of the results presented here have been previously discussed in [47,55,56,57,58].
Softky and Koch  have recently pointed out a puzzling conflict between the characteristics of spike trains recorded from cortical cells responding at high rates to visual input and standard biophysical theories. Experimental evidence shows that the amplitude of an individual excitatory postsynaptic potential (EPSP) is on the order of 0.1 mV, about two orders of magnitude smaller than the threshold depolarization from rest necessary for a pyramidal cell to spike [25,30]; for a review see also . Based on this,  showed that the neural firing pattern will be highly regular if the neuronal membrane acts as a leaky integrator summing over a train of stochastic, uncorrelated EPSP's. In an integrator model, the time to spike is determined by the total time in which a critical number of EPSP's accumulate. Since the interspike interval is the sum of random variables representing the intervals between EPSP inputs, the central limit theorem predicts that the output spikes will be highly regular. In other words, the shape of the interspike interval histogram will become highly peaked as measured by the coefficient of variation, , defined as the standard deviation over the mean of the Interspike Interval (ISI) distribution.  also showed that this central limit result holds for a detailed biophysical compartmental model (including seven voltage-dependent somatic currents) of a cortical pyramidal cell in the presence of independent synaptic input. However, recordings from cells in V1 and MT cortex in the behaving monkey show that the discharge at high rates (up to 200 Hz) is highly variable in the length of interspike intervals with a coefficient of variation of around one.
Despite the irregularity of single spike trains, many groups report high-frequency oscillations in visual and sensory-motor cortex [10,11,15,26,27,32]. In general these oscillations are more visible in global activity variables, such as local field potential (LFP) and multi-unit recordings (MUA), than in the spiking activity from single cells which typically are not oscillatory .
Taken together, these results pose a challenge for neural modeling: what kind of dynamic process can simultaneously generate oscillations in local field variables and irregular discharge patterns without oscillatory peaks in the power spectrum at the single cell level? Surely the naive but widely used assumption that spike trains can be treated as Poisson distributed point processes, while explaining the broad interval distribution found in single cells, are completely at a loss to explain the robust oscillations in local field variables.
We will present here a model of leaky integrate-and-fire units embedded into a simple neuronal network with local excitation and surround inhibition to explain both observations. We then address the function played by this system in visual information processing. In order to do this, we add to the general cortical model precise constraints on the connectivity patterns of horizontal projections measured in cortex [14,28,29], and take into consideration the map of V1 cells' orientation preferences [6,7]). Models of cortical activity built upon those constrains are then compared to experimental results in physiology [17,23,34] and in psychophyscs [40,41] which address the nature of lateral interaction in visual processing. We will show that the horizontal connections can play an essential role in three types of computations: binding and grouping by synchronized LFP oscillations, pop-out and line completion.