Spike-based neuromorphic learning circuits typically represent their synaptic weights as voltages, and convert them into post-synaptic currents so that they can be integrated by their afferent silicon neuron. This voltage-to-current conversion is often done using a single transistor. This results in an exponential (for weak-inversion) or quadratic (for strong inversion) non-linear transformation which severely restricts the type of learning algorithms that can be implemented. To overcome this problem we propose a range of solutions that perform a linear transformation fro m weight voltage to synaptic current, simplifying the implementation of a spike-based learning rules. We demonstrate the application of these conversion circuits using current-mode integrators that produce alpha-functions with biologically realistic temporal dynamics and amplitudes that are linearly proportional to the synaptic weights. The circuits proposed are low-power, and can be integrated in a wide range of spike-based learning framework s that have been recently proposed. We describe the advantages and disadvantages of the various solutions proposed and validate them with circuit simulation results.