Owing to neurotechnological advances in electrode design, it is now possible to simultaneously record spiking activity from hundreds to thousands of neurons. Such extensive data provides an opportunity to study how groups of neurons coordinate to form functional ensembles that ultimately drive behavior. Since the spike train space is devoid of an algebraic structure, quantifying causal relations between the neuronal nodes poses a computational challenge. Here, we combine techniques from information theory and kernel-based spike train representations to construct an estimator of directed information for causal analysis between neural spike train data. Via projection of spiking data into a reproducing kernel Hilbert space, we avoid tedious evaluations of probability distribution while engaging computations in a non-linear space of (possibly) infinite dimensionality. Additionally, the estimator allows for conditioning on `side' variables to eliminate indirect causal influences in a multi-neuron network. Extensive analyses on a simulated six-neuron network model comprising of different neuron types and causal topologies show that the devised measure identifies directional influences accurately that would be otherwise inaccessible with traditional correlation measures. Finally, we apply the metric to identify direct causal interactions among neurons recorded from cortical columns of visual-area 4 of monkeys performing a delayed match to sample task. Our results reveal an interesting reorganization of neuronal interaction patterns within a cortical column on visual stimulation.