No abstract

Grid cell responses develop gradually after eye opening, but little is known about the rules that govern this process. We present a biologically plausible model for the formation of a grid cell network. An asymmetric spike time-dependent plasticity rule acts upon an initially unstructured network of spiking neurons that receive inputs encoding animal velocity and location. Neurons develop an organized recurrent architecture based on the similarity of their inputs, interacting through inhibitory interneurons. The mature network can convert velocity inputs into estimates of animal location, showing that spatially periodic responses and the capacity of path integration can arise through synaptic plasticity, acting on inputs that display neither. The model provides numerous predictions about the necessity of spatial exploration for grid cell development, network topography, the maturation of velocity tuning and neural correlations, the abrupt transition to stable patterned responses, and possible mechanisms to set grid period across grid modules.

In the original version of this paper published online, an incorrect version of Figure 7 was used as the result of a production error at Cell Press (the symbol legend in panel C was incorrect). We have corrected the figure online as well as in the print issue, and the journal regrets the error.

This Letter presents a quantitative theory of resonant mixing in time-dependent volume-preserving 3D flows using a model cellular flow introduced in T. Solomon and I. Mezic, Nature (London) 425, 376 (2003), as an example. Specifically, we show that chaotic advection is dramatically enhanced by a time-dependent perturbation for certain resonant frequencies. We compute the fraction of the mixed volume as a function of the frequency of the perturbation and show that essentially complete mixing in 3D is achieved at every resonant frequency.

We consider mixing via chaotic advection in microdroplets suspended at the free surface of a liquid substrate and driven along a straight line using the thermocapillary effect. With the help of a model derived by Grigoriev ͓Phys. Fluids 17, 033601 ͑2005͔͒ we show that the mixing properties of the flow inside the droplet can vary dramatically as a function of the physical properties of the fluids and the imposed temperature profile. Proper characterization of the mixing quality requires introduction of two different metrics. The first metric determines the relative volumes of the domain of chaotic streamlines and the domain of regular streamlines. The second metric describes the time for homogenization inside the chaotic domain. We compute both metrics using perturbation theory in the limit of weak temperature dependence of the surface tension coefficient at the free surface of the substrate.