We have developed a digital holographic microscope (DHM), in a transmission mode, adapted to the quantitative study of cellular dynamics. Living cells in culture are optically probed by measuring the phase shift they produce on the transmitted wave front. The high temporal stability of the phase signal, equivalent to lambda/1800, and the low acquisition time (~20micros) enable to monitor cellular dynamics processes. An experimental procedure allowing to calculate both the integral refractive index and the cellular thickness (morphometry) from the measured phase shift is presented. Specifically, the method has been applied to study the dynamics of neurons in culture during a hypotonic stress. Such stress produces a paradoxical decrease of the phase which can be entirely resolved by applying the methodological approach described in this article; indeed the method allows to determine independently the thickness and the integral refractive index of cells.