The standard two-step scheme for modeling extracellular signals is to first compute the neural membrane currents using multicompartment neuron models (step 1) and next use volume-conductor theory to compute the extracellular potential resulting from these membrane currents (step 2). In this chapter, we introduce ways to implement this scheme in computer simulations based on designated software such as LFPy, the NEURON simulator, or the Arbor simulator. We also introduce various methods for reducing the computational cost of simulating the extracellular potentials of large networks of neurons as well as introduce heuristic approximate signal prediction methods.

Kernel methods provide an alternative family of non-linear methods to neural networks, with support vector machine being the best known among kernel methods. Almost all linear statistical methods have been non-linearly generalized by the kernel approach, including ridge regression, linear discriminant analysis, principal component analysis, canonical correlation analysis, and so on. The kernel method has also been extended to probabilisitic models, for example Gaussian processes.

Starting from this chapter, Part III introduces several commonly used algorithms in pattern recognition and machine learning. Support vector machines (SVM) starts from a simple and beautiful idea: large margin. We first show that in order to find such an idea, we may need to simplify our problem setup by assuming a linearly separable binary one. Then we visualize and calculate the margin to reach the SVM formulation, which is complex and difficult to optimize. We practice the simplification procedure again until the formulation becomes viable, briefly mention the primal--dual relationship, but do not go into details of its optimization. We show that the simplification assumptions (linear, separable, and binary) can be relaxed such that SVM will solve more difficult tasks---and the key ideas here are also useful in other tasks: slack variables and kernel methods.