Factors affecting phase synchronization in integrate-and-fire oscillators

The dynamics of integrate-and-fire: mean versus variance modulations and dependence on baseline parameters

The leaky integrate-and-fire (LIF) is the simplest neuron model that captures the essential properties of neuronal signaling. Yet common intuitions are inadequate to explain basic properties of LIF responses to sinusoidal modulations of the input. Here we examine responses to low and moderate frequency modulations of both the mean and variance of the input current and quantify how these responses depend on baseline parameters. Across parameters, responses to modulations in the mean current are low pass, approaching zero in the limit of high frequencies. For very low baseline firing rates, the response cutoff frequency matches that expected from membrane integration. However, the cutoff shows a rapid, supralinear increase with firing rate, with a steeper increase in the case of lower noise. For modulations of the input variance, the gain at high frequency remains finite. Here, we show that the low-frequency responses depend strongly on baseline parameters and derive an analytic condition specifying the parameters at which responses switch from being dominated by low versus high frequencies. Additionally, we show that the resonant responses for variance modulations have properties not expected for common oscillatory resonances: they peak at frequencies higher than the baseline firing rate and persist when oscillatory spiking is disrupted by high noise. Finally, the responses to mean and variance modulations are shown to have a complementary dependence on baseline parameters at higher frequencies, resulting in responses to modulations of Poisson input rates that are independent of baseline input statistics.

Complementary responses to mean and variance modulations in the perfect integrate-and-fire model

In the perfect integrate-and-fire model (PIF), the membrane voltage is proportional to the integral of the input current since the time of the previous spike. It has been shown that the firing rate within a noise free ensemble of PIF neurons responds instantaneously to dynamic changes in the input current, whereas in the presence of white noise, model neurons preferentially pass low frequency modulations of the mean current. Here, we prove that when the input variance is perturbed while holding the mean current constant, the PIF responds preferentially to high frequency modulations. Moreover, the linear filters for mean and variance modulations are complementary, adding exactly to one. Since changes in the rate of Poisson distributed inputs lead to proportional changes in the mean and variance, these results imply that an ensemble of PIF neurons transmits a perfect replica of the time-varying input rate for Poisson distributed input. A more general argument shows that this property holds for any signal leading to proportional changes in the mean and variance of the input current.

Neuronal functional diversity and collective behaviors: a scientific case

A major issue in today's neuroscience is how the brain complex and highly flexible organization emerges from its individual components. Robustness of neuronal properties with weak linkages between regulatory processes are suggested to account for the adaptive, tunable, multistable dynamics, the coding schemes and the complexity of neuronal functional (sub)systems. Interneurons and neurotransmitter diversity, resonance phenomena due to properties of the cell or network, time/frequency-dependent activation of dedicated neuronal assemblies, code- and frequency-specific oscillations interact in determining the brain functional setup and operations. Despite the scientific relevance, comprehensive theories are not yet available, but the scenario--however incomplete and incompletely characterized--is promising and warrants further investigation.

Neuronal functional diversity and collective behaviors

A major question in today's neuroscience is how the brain's complex operations and organization emerge from individual components. The robustness of neuronal properties with flexible linkages between regulatory processes conceivably accounts for the adaptive, tunable, multistable dynamics; the coding schemes; and the complexity of neuronal functional (sub)systems. Interneurons and neurotransmitter diversity, resonance phenomena due to properties of the cell, time/frequency-dependent activation of dedicated neuronal assemblies, and code- and frequency-specific oscillations interact in determining the brain functional setup and operations. Such an arrangement would also provide the functional requirements for access to neural mechanisms, dedicated neuronal circuitry and the proper timing allowing for the selective differentiation among cortical neurons due to performing in different tasks. No comprehensive theory or systematic methodological approach appears yet conceivable. The scenario, however incomplete and incompletely characterized, is nevertheless promising and warrants further investigation.