Neuromorphic processors that implement Spiking Neural Networks (SNNs) using mixed-signal analog/digital circuits represent a promising technology for closed-loop real-time processing of biosignals. As in biology, to minimize power consumption, the silicon neurons' circuits are configured to fire with a limited dynamic range and with maximum firing rates restricted to a few tens or hundreds of Herz. However, biosignals can have a very large dynamic range, so encoding them into spikes without saturating the neuron outputs represents an open challenge. In this work, we present a biologically-inspired strategy for compressing this high-dynamic range in SNN architectures, using three adaptation mechanisms ubiquitous in the brain: spike-frequency adaptation at the single neuron level, feed-forward inhibitory connections from neurons belonging to the input layer, and Excitatory-Inhibitory (E-I) balance via recurrent inhibition among neurons in the output layer. We apply this strategy to input biosignals encoded using both an asynchronous delta modulation method and an energy-based pulse-frequency modulation method. We validate this approach in silico, simulating a simple network applied to a gesture classification task from surface EMG recordings.