autores contribuíram igualmente na escrita e elaboração do presente trabalho.Físicas e físicos têm começado a trabalhar em áreas onde é necessária a análise de sinais ruidosos. Nessas áreas, tais como a Economia, a Neurociência e a Física, a noção de causalidade deve ser interpretada como uma medida estatística. Introduzimos ao leitor leigo a causalidade de Granger entre duas séries temporais e ilustramos como calculá-la: um sinal X "Granger-causa" Y se a observação do passado de X aumenta a previsibilidade do futuro de Y em comparação com o que é possível prever apenas pela observação do passado de Y . Em outras palavras, para haver causalidade de Granger entre dois sinais basta que a informação do passado de um melhore a previsão do futuro de outro, mesmo na ausência de mecanismos físicos de interação. Apresentamos a derivação da causalidade de Granger nos domínios do tempo e da frequência e damos exemplos numéricos através de um método nãoparamétrico no domínio da frequência. Métodos paramétricos são abordados no Apêndice. Discutimos limitações e aplicações desse método e outras alternativas para medir causalidade. Palavras-chave: Causalidade de Granger, processo autoregressivo, causalidade de Granger condicional, estimação não-paramétrica Physicists are starting to work in areas where noisy signal analysis is required. In these fields, such as Economics, Neuroscience, and Physics, the notion of causality should be interpreted as a statistical measure. We introduce to the lay reader the Granger causality between two time series and illustrate ways of calculating it: a signal X "Granger-causes" a signal Y if the observation of the past of X increases the predictability of the future of Y when compared to the same prediction done with the past of Y alone. In other words, for Granger causality between two quantities it suffices that information extracted from the past of one of them improves the forecast of the future of the other, even in the absence of any physical mechanism of interaction. We present derivations of the Granger causality measure in the time and frequency domains and give numerical examples using a non-parametric estimation method in the frequency domain. Parametric methods are addressed in the Appendix. We discuss the limitations and applications of this method and other alternatives to measure causality.
The conventional impedance profile of a neuron can identify the presence of resonance and other properties of the neuronal response to oscillatory inputs, such as nonlinear response amplifications, but it cannot distinguish other nonlinear properties such as asymmetries in the shape of the voltage response envelope. Experimental observations have shown that the response of neurons to oscillatory inputs preferentially enhances either the upper or lower part of the voltage envelope in different frequency bands. These asymmetric voltage responses arise in a neuron model when it is submitted to high enough amplitude oscillatory currents of variable frequencies. We show how the nonlinearities associated to different ionic currents or present in the model as captured by its voltage equation lead to asymmetrical response and how high amplitude oscillatory currents emphasize this response. We propose a geometrical explanation for the phenomenon where asymmetries result not only from nonlinearities in their activation curves but also from nonlinearites captured by the nullclines in the phase-plane diagram and from the system's time-scale separation. In addition, we identify an unexpected frequency-dependent pattern which develops in the gating variables of these currents and is a product of strong nonlinearities in the system as we show by controlling such behavior by manipulating the activation curve parameters. The results reported in this paper shed light on the ionic mechanisms by which brain embedded neurons process oscillatory information.
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