  Homeostatic plasticity is a stabilizing mechanism commonly observed in real
neural systems that allows neurons to maintain their activity around a
functional operating point. This phenomenon can be used in neuromorphic systems
to compensate for slowly changing conditions or chronic shifts in the system
configuration. However, to avoid interference with other adaptation or learning
processes active in the neuromorphic system, it is important that the
homeostatic plasticity mechanism operates on time scales that are much longer
than conventional synaptic plasticity ones. In this paper we present an
ultra-low leakage circuit, integrated into an automatic gain control scheme,
that can implement the synaptic scaling homeostatic process over extremely long
time scales. Synaptic scaling consists in globally scaling the synaptic weights
of all synapses impinging onto a neuron maintaining their relative differences,
to preserve the effects of learning. The scheme we propose controls the global
gain of analog log-domain synapse circuits to keep the neuron's average firing
rate constant around a set operating point, over extremely long time scales. To
validate the proposed scheme, we implemented the ultra-low leakage synaptic
scaling homeostatic plasticity circuit in a standard 0.18 $\mu$m Complementary
Metal-Oxide Semiconductor (CMOS) process, and integrated it in an array of
dynamic synapses connected to an adaptive integrate and fire neuron. The
circuit occupies a silicon area of 84 $\mu$m x 22 $\mu$m and consumes
approximately 10.8 nW with a 1.8 V supply voltage. We present experimental
results from the homeostatic circuit and demonstrate how it can be configured
to exhibit time scales of up to 100 kilo-seconds, thanks to a controllable
leakage current that can be scaled down to 0.45 atto-Amperes (2.8 electrons/s).
