We study the Hankel determinant generated by the Gaussian weight with jump discontinuities at t1,…, tm. By making use of a pair of ladder operators satisfied by the associated monic orthogonal polynomials and three supplementary conditions, we show that the logarithmic derivative of the Hankel determinant satisfies a second-order partial differential equation which is reduced to the σ-form of a Painlevé IV equation when m = 1. Moreover, under the assumption that tk − t1 is fixed for k = 2,…, m, by considering the Riemann-Hilbert problem for the orthogonal polynomials, we construct direct relationships between the auxiliary quantities introduced in the ladder operators and solutions of a coupled Painlevé IV system.