No abstract

Amorphous packings of nonspherical particles such as ellipsoids and spherocylinders are known to be hypostatic: The number of mechanical contacts between particles is smaller than the number of degrees of freedom, thus violating Maxwell’s mechanical stability criterion. In this work, we propose a general theory of hypostatic amorphous packings and the associated jamming transition. First, we show that many systems fall into a same universality class. As an example, we explicitly map ellipsoids into a system of “breathing” particles. We show by using a marginal stability argument that in both cases jammed packings are hypostatic and that the critical exponents related to the contact number and the vibrational density of states are the same. Furthermore, we introduce a generalized perceptron model which can be solved analytically by the replica method. The analytical solution predicts critical exponents in the same hypostatic jamming universality class. Our analysis further reveals that the force and gap distributions of hypostatic jamming do not show power-law behavior, in marked contrast to the isostatic jamming of spherical particles. Finally, we confirm our theoretical predictions by numerical simulations.

1. Post‐stimulus histograms were obtained from ‘sustained’ and ‘transient’ retinal ganglion cells for receptive field plots using a light spot with square‐wave modulation of intensity, and of variable intensity and area. Fundamental differences in their receptive field organization in time and space were revealed. 2. In ‘sustained’ cells, excitation consists of ‘transient’ and ‘sustained’ components and the ratio of transient/sustained components remains constant at a given retinal locus for a wide range of intensities. The transient component becomes proportionally larger towards the periphery of the receptive field. This rule is also applicable for the inhibitory and disinhibitory surround. In ‘transient’ cells, however, there is no true ‘sustained’ component, but some cells produce a double peaked transient post‐stimulus histogram at the R.F. centre when high flux stimuli are used, while others show a single peak transient response. The magnitude and shape of transient responses changes with intensity as well as with location in the receptive field. 3. The sensitivity gradients of ‘sustained’ and ‘transient’ cells show consistent differences in shape. The mean slope of the sensitivity gradients of a sample of ‘sustained’ cells was 10 times that of a sample of ‘transient’ cells. The sensitivity gradient of ‘sustained’ cells shows a distinct surround region where the inhibitory mechanism is more sensitive, while that of ‘transient’ cells usually does not, owing to an extensive ‘tail’ on the sensitivity gradient of the centre mechanism, which overlaps the surround. 4. Ricco's Law also holds for the centre mechanism of ‘transient’ cells. Non‐linear summation occurs at supra‐threshold levels, and when the surround mechanisms are involved. 5. Supra‐optimal stimuli give a saturation of the response in both ‘transient and ‘sustained’ cells. This saturation is associated with a decrease of latency in ‘transient’ cells, but not in ‘sustained’ cells. 6. The latency of retinal ganglion cells is determined by both stimulus and background flux. The effect of the background is negligible except at low values of stimulus flux, where its effect may be analysed primarily in terms of its effect on the incremental threshold. 7. The latency to stimulation with a standard small spot (25–27′) at the receptive field centre is shorter for ‘sustained’ cells than for ‘transient’ cells; this latency difference being related to the greater sensitivity of the ‘sustained’ cells to stimuli of this size. Differences in conduction time along ‘transient’ and ‘sustained’ pathways to the lateral geniculate nucleus (LGN) and cortex were estimated, and it is concluded that despite the latency difference noted above, a response to a stimulus which is optimal for a ‘transient’ cell reaches the cortex faster than the response to a stimulus which is optimal for a ‘sustained’ cell. 8. The above results together with previous evidence available suggest that for most stimuli, centre and surround mechanisms are activated simultaneously and algebraically su...