We present a fast algorithm for the resolution of the Lasso for convolutional models in high dimension, with a particular focus on the problem of spike sorting in neuroscience. Making use of biological properties related to neurons, we explain how the particular structure of the problem allows several optimizations, leading to an algorithm with a temporal complexity which grows linearly with respect to the size of the recorded signal and can be performed online. Moreover the spatial separability of the initial problem allows to break it into subproblems, further reducing the complexity and making possible its application on the latest recording devices which comprise a large number of sensors. We provide several mathematical results: the size and numerical complexity of the subproblems can be estimated mathematically by using percolation theory. We also show under reasonable assumptions that the Lasso estimator retrieves the true support with large probability. Finally the theoretical time complexity of the algorithm is given. Numerical simulations are also provided in order to illustrate the efficiency of our approach.