abstract

We simulate a recurrent neural network composed of 100 neurons based on the leaky integrate-and-fire model where there are both excitatory and inhibitory connections. When the neurons in the network are driven by a constant external current and when the global inhibition dominates, stable periodic spiking patterns form after a short transient period. We then inject noise into the system by adding excitatory synapses which fire according to a Poisson process. We vary both the noise strength and the noisy synapse firing rate and record how this affects the periodic spiking pattern.

Recurrent neural networks are groups of neurons connected by excitatory and inhibitory synapses. In general, any neuron may connect to any other neuron, and thus feedback is possible. These networks have often been used to model computational processes that occur in the brain. More recently, stable periodic spiking patterns which emerge when the network is driven have been studied as a possible model for short-term memories. We simulate a very general recurrent neural network, and use it to study the effects of noise on periodic spiking patterns.

Our recurrent neural network is modeled as a set of pulse-coupled leaky integrate-and-fire (LINF) neurons. The neurons are connected by both inhibitory and excitatory synapses whose strengths are randomly assigned. These strengths are described by the conductance matrices G^E_ij for the excitatory synapses and G^I_ij for the inhibitory synapses. However, it is stipulated that the global inhibition of the network must dominate so that two neurons may not fire at the same time. All neurons are driven by a constant external current. We simulated 100 neurons, and observed the spiking patterns. A portion of our network is displayed to the right. Excitatory connections are in red, inhibitory connections are in blue.

Mathematically, the LINF neurons are described by the equations:

where &tau is the membrane time constant, V_j is the membrane potential of the j-th neuron, V_rest is the resting membrane potential, I_j is the external current applied to the j-th neuron, I_exc is the effective current from the excitatory synapses, and I_ihb is the effective current from the inhibitory synapses. More specifically,

where g_j is the effective conductance of all the excitatory synapses connected to the j-th neuron. Likewise,

where g_j is the effective conductance of all the inhibitory synapses connected to the j-th neuron. Note that the reversal potential of the excitatory connections is 0mV while the reversal potential of the inhibitory connections is V_ihb. The effective conductances are determined using the G_ij conductance matrix. When the i-th neuron fires:

When no neurons are firing, the conductance decays exponentially:

It has been shown, under very general conditions, that the spike sequence of such models converge to stable periodic patterns [1]. Furthermore, these periodic patterns are thought to underlie various computational processes in the brain such as short-term memory. Our simulated network of 100 neurons again demonstrates this result. After a short transient period, our network converged to the non-trivial periodic spiking pattern displayed to the right. Here, we only show 16 of the 100 neurons because only 16 neurons ever fired. Also, it is important to note that the transient period is short due to the dominance of global inhibition.

There are still many unresolved questions relating to these spiking patterns. Namely, how does changing the neuron model affect the spiking pattern, and how does injecting noise into the system affect the spiking pattern? We examine the effects of noise.

We modeled noise by connecting an excitatory synapse to each neuron in the network that fires randomly according to a Poisson process. Effectively, we are adding a term I_noise to dynamical equation governing the neurons. The equations governing I_noise are exactly analogous to those governing the excitatory connections, except there is no conductance matrix for noise. Instead, the effective conductance g_j increases by some fixed value, the noise strength, whenever the excitatory connection randomly fires.

Thus, there are two parameters of the noise that we may vary to analyze how the noise affects the periodic patterns, the strength of the noise and the firing rates of the noise synapses.

For a small amount of noise, there was no effect on the spiking pattern. Here, we show the same spiking pattern along with the noise spikes in red. However, as we increase the noise, eventually the pattern is destroyed or possibly changed to a different spiking pattern.

We varied the noise strength from 0 mV to 20 mV and the noise spiking rate from 0 Hz to 2 Hz, and we recorded whether a pattern occurred. If a pattern did occur, we recorded the duration of the transient period before the pattern emerged. The results are displayed in the graph to the right. Warmer colors represent patterns that started later. Blue indicates that a pattern never formed.

Clearly, increased noise, measured by both noise strength and noise spiking rate, leads to a decrease in the probability of forming a stable periodic pattern. However, even small amounts of noise can destroy a pattern. This leads us to believe that single, appropriately positioned spikes can destroy a spiking pattern, but that generally the patterns are robust against small amounts of noise.