The membrane potential never reaches steady state, so the time constant is irrelevant. Firing rate is better described by current than voltage. Applying this principle to the dynamics of simple feedback networks shows that a key factor in the response time of a network is the adaptation current. Without adaptation, the network time constant can be long because it is the gain of the network multiplied by the synaptic time constants. Adaptation can cancel out the slow component of synaptic current, significantly speeding up response times.
Recurrent inhibition has a similar effect.
Another key factor determining input current is synaptic depression and facilitation. However, facilitation and depression can act together to provide a log-exponential transform, allowing subtractive inhibition at one stage to have a divisive effect at another.