We present a scheme for implementing highly-connected, reconfigurable networks of integrate-and-fire neurons in VLSI. Neural activity is encoded by spikes, where the address of an active neuron is communicated through an asynchronous request and acknowledgement cycle. We employ probabilistic transmission of spikes to implement continuous-valued synaptic weights, and memory-based look-up tables to implement arbitrary interconnection topologies. The scheme is modular and scalable, and lends itself to the implementation of multi-chip network architectures. Results from a prototype system with 1024 analog VLSI integrate-and-fire neurons, each with up to 128 probabilistic synapses, demonstrate these concepts in an image processing task.