To understand the functioning of a complex system, it is often useful to develop a map of interactions between the system's components. This “network science” approach has been applied to a wide variety of systems with great success 
Typically, interactions between components are classified as physical, functional, or effective 
. In neuroscience, a synapse or gap junction would constitute a physical connection between neurons; a correlation between spike trains of two neurons would constitute a functional connection; the ability to predict the firing of one neuron based on the firing of another neuron would constitute an effective connection.
Physical connections delimit the ways in which activity could
flow within a circuit, whereas effective connections describe the ways in which activity typically
flows. Note that effective connections can be a small subset of physical connections. Thus, knowledge of effective connectivity may provide insights into how information is typically distributed and recombined in neural circuits. An effective connectivity map would permit many powerful graph-theoretic tools to be applied 
, potentially allowing subtle differences in information processing between healthy and diseased networks to be identified. Now that it is possible to record activity from hundreds of closely spaced neurons at high temporal resolution for several hours at a time 
, it is extremely important to have an accurate and robust measure of effective connectivity between neurons.
In this regard, transfer entropy (TE) has recently received much attention in neuroscience 
. It is an information-theoretic measure first introduced by Thomas Schreiber 
to assess effective connectivity. As an information-theoretic measure, it has often been claimed that TE can be used to estimate “information flow” between neurons 
. In a recent paper that surveyed several different information theoretic methods 
, the authors concluded that TE performed better at identifying effective connectivity in complex systems than cross-correlation (CC), mutual information, and joint entropy. Another recent study compared TE to extended and nonlinear Granger causality, and to predictability improvement. These authors concluded that TE was superior in terms of stability and accuracy when applied to time series from linear and non-linear models with unidirectional and bidirectional coupling 
. Thus, although there are many potential ways to assess interactions between neurons including Granger causality 
, the directed transfer function (DTF) 
, generalized linear models 
, Bayesian models 
, data mining techniques 
, partial directed coherence 
, and many others 
, TE is gaining wide acceptance.
Despite the accuracy of TE, it has several limitations. First, most of the published work using TE has assessed interactions at only one time delay 
, but see 
. This is problematic because delays between an action potential and a post-synaptic potential typically range from one to twenty milliseconds in the mammalian cortex 
. To identify such connections with single-delay TE (D1TE), one might try to use large time bins (~20 ms), but this could sacrifice temporal resolution. This suggests a need to investigate TE with multiple delays. Second, all of the authors to our knowledge have only considered messages between neurons of a single bin length. While such measures would capture information sent between neurons in the form of a single spike, they would exclude messages spanning multiple time bins, like bursts. As bursts are widely hypothesized to be an important type of message sent between neurons 
, it would be desirable to measure TE with message lengths beyond one bin. To resolve these problems, we here introduce an extension to TE that allows us to probe connectivity at multiple time delays (due, for example, to synaptic connections or axonal signal propagation), selecting the peak value of TE. We call this delayed TE. Furthermore, we also extend Delayed TE by considering message lengths up to 5 bins long at multiple delays. We call this higher-order TE (HOTE). We offer these extensions as freeware on our project website (http://code.google.com/p/transfer-entropy-toolbox/
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To evaluate these extensions to TE (delayed TE, HOTE), we wanted to compare them to D1TE and CC. We selected CC because, in its time-lagged form, it has been widely used to assess effective connectivity between neurons 
. The CC does not produce a single number, but a set of numbers indicating the strength of a connection at various delays. It is thus necessary to develop a method for comparing CC curves. Interestingly, we found that there was not complete agreement in the literature on this topic. In particular, some research papers did not specify a method for normalizing the CC, even though this can substantially affect results. To alleviate this situation, we adopt a standard coincidence index (CI) 
. The CI allows the peak region of a measurement to be selected when it is assessed at many delays. Accordingly, we applied the CI to delayed TE, HOTE and to CC.
As our eventual goal is to assess effective connectivity in experimental data, we applied our methods to a neural network model as a first step in validating our approach. Izhikevich's cortical network model 
is widely used as it is computationally inexpensive and has many realistic features including cell types with different intrinsic firing patterns 
, different synaptic delays, and spike-timing dependent plasticity (STDP) 
. In addition, this model is known to capture several emergent properties present in cortical circuits, including gamma oscillations 
and repeating patterns of spike activity (polychronous groups) 
. These emergent phenomena pose severe challenges for methods of assessing effective connectivity, as they frequently produce situations in which many neurons appear to be driving another neuron later in time. We applied our methods to Izhikevich networks containing 1000 neurons. Previous groups have validated their methods in simplified circuits containing relatively few model neurons 
, on a few neurons embedded in a larger network 
, or in networks with only a single time delay 
, so the task posed here is challenging. By using a model with known connectivity, we were able to measure the true positive and false positive rates of each method, allowing for objective comparisons of performance.