correlated with process of the source neuron and was uncorrelated with that of the target neuron. The GCI decreased because the effect of ξ2 dominated the effect of ξ3 and ξ4. In other words, the error process was composed of spikes of some connected neurons which were positively correlated with the source neuron.
FP-increase: The GCIs of neuron pairs (16 of 30 pairs) increased as the error percentage increased under the FP-procedure (Figure 2.7(a)) and the four corre-sponding ξ’s of this pattern (Figure 2.7(d)) show that the error process was posi-tively correlated with processes of both the source and target neurons. The GCI increased because the effect of ξ3 and ξ4 dominated the effect of ξ2. In other words, the error process was composed of spikes of some connected neurons which were strongly correlated with the target neuron.
Finally, we note that there are 5 neuron pairs with a zero GCI, and 3 neuron pairs with unchanged GCIs under the FP-procedure.
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0.0 0.2 0.4 0.6 0.8
0.000.050.100.15
(a) Results of real data
Error Percentage r
GCI
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FN−decrease FP−decrease FP−increase
0.0 0.2 0.4 0.6 0.8
−4−20246
(b) The ξ1 to ξ4 of FN−decrease
Error Percentage r ξ1ξ2
ξ3ξ4
0.0 0.2 0.4 0.6 0.8
02468
(c) The ξ1 to ξ4 of FP−decrease
Error Percentage r ξ1ξ2
ξ3ξ4
0.0 0.2 0.4 0.6 0.8
02468
(d) The ξ1 to ξ4 of FP−increase
Error Percentage r ξ1ξ2
ξ3ξ4
Figure 2.7: (a) Three kinds of Granger Causality Index (GCI) patterns which frequently appeared in real experimental data. (b) ξ1 to ξ4 of the false-negative (FN)-decrease pattern. (c) ξ1 to ξ4 of the false-positive (FP)-decrease pattern. (d) ξ1 to ξ4 of the FP-increase pattern.
this is a relatively conservative and secure strategy for scientific research. For (i) and (ii), in fact, we really cannot avoid missing spikes successively or adding spikes concentratively in the analysis of real data. However, our suggestions still are useful in some cases when recordings are made in the brain regions that neurons are known with complementary intermittence discharge. For example, since the inspiratory-related and expiratory-related neurons coexist in the dorsal and ventral respiratory group [65], and the firing of these two types of neurons are complementary intermittence, so successive missing or concentrative adding of spikes after sorting may occur. We should examine the time series of spike trains
to ensure if the patterns of complementary intermittence are confused after sorting, and to infer which modification would be made when GCIs are calculated. For (iii) and (iv), these two conclusions are just opposite to each other. That is, in spike detection, the FPs are better than the FNs, because the FPs consist of only electrical noises. But in spike classification, the FNs are better than the FPs because the FPs consist of not only electrical noises but also maybe some causal neurons. Finally we note that the way of choosing an optimal threshold or cluster size varies from case to case, since it depends on the sorting method used, and the experimental situation you met. This study is just trying to give a general concept for choosing a better threshold and cluster size.
The results of this study are based on restrictive situations. The analytic formula was obtained from a first order autoregressive model, and the error processes were only superposed on the source process. However, based on these simplifications, the intrinsic properties of the GCI can be seen more clearly than in complete but more complicated situations. Although there are still a lot of concerns on the technical aspect of applying the GCI to determine the relationship among neurons in practice, researchers may be interested in understanding intuitively the effect of spike sorting error before these tech-niques are really applied, and this is exactly what this chapter wants to provide. Real neuronal networks are much more complex than the simplified assumptions of the anal-yses and simple models of the simulation. The procedures presented in this study need further development to approach the complex reality. The well-established framework of information theory, for example, might be employed to provide more-credible statistical inferences about true causality in the future.
Chapter 3
Synaptic Weights Estimation
3.1 Introduction
Granger causality (GC) [27, 28] has been shown to be an effective method for analyzing the causal relationship between continuous-valued neural activity data [3,8,13,18] and has been widely deployed in recent neuroscience research. To further understand how neurons cooperate to generate specific brain functions, several extended GC methods were also proposed for identifying directional interactions between neurons through multiple spike trains [38, 41, 51, 57, 74]. Being the fundamental knowledge used in this chapter, the time domain GC analysis will be briefly introduced in the next section and the readers are referred to an article by Barnett and Seth [4] for more details.
The term synaptic weight is widely used in neural network research and typically refer to the coupling strength of a connection between two nodes in the network. A large synaptic weight usually means that a large signal (i.e., high-frequency spikes) from the pre-synaptic neuron can result in a large signal of the post-synaptic one. Therefore, in neuroscience and biology, it can also be interpreted as the amount of influence of one neuron has on the firing activity of another.
The spikes of a pre-synaptic neuron are carried by the axon, which will release exci-tatory or inhibitory neurotransmitter into the synapse. When the post-synaptic neuron receives the neurotransmitter, an excitatory post-synaptic potential (EPSP) or an in-hibitory post-synaptic potential (IPSP) is then induced to temporarily depolarize or hy-perpolarize the membrane potential. An EPSP makes the neuron more likely to generate an action potential (AP), while an IPSP makes the neuron to do the opposite. However, a single EPSP is not sufficient for the membrane to generate an AP, temporal or spatial summations are required. This means that the firing pattern of the post-synaptic neuron is generally not a direct consequence of the influence of a single pre-synaptic neuron; but a weighted result of the effects of several pre-synaptic neurons with possibly different synaptic weights. Furthermore, IPSPs will diminish EPSPs, playing a much more cru-cial role of determining whether or not an AP generation will occur at the post-synaptic membrane.
The GC analysis has emerged as a powerful analytical method for estimating the
causal strength of complex networks [58, 59]. However, the effects of excitations and inhibitions could not be differentiated in its original form. Based on the GC, we pro-pose a computational algorithm (presented in Section 3.2.3) under the assumption of best linear predictor (BLP) for analyzing neuronal networks by estimating the synaptic weights among them. The idea of the mathematical assumption BLP is that the weighted voltage-fluctuation of the pre-synaptic neurons should be the best linear explanation for the voltage-fluctuation of the post-synaptic neuron among the network. Using this inter-pretation, the GC analysis can be extended to measure both excitatory and inhibitory effects between neurons without too much extra computational complexity. The appro-priateness of the BLP assumption was examined by three sorts of simulated networks:
those with linear, almost linear, and nonlinear network structures. To illustrate the ap-plication of the proposed method, real spike trains from the anterior cingulate cortex (ACC) and the striatum (STR) were analyzed.
It is worth noting that spike trains are non-equally spaced data and are regarded as being from a point process. Filtering is usually required for converting them to equally spaced time series for further GC analyses [57]. This study adopted the Gaussian kernel filtering or binning (depending on the situation) to convert spike trains into time series data for the following three main reasons: (1) it reduces the complexity of analysis, and considers also the effect of temporal summation of action potentials, (2) under suitable preprocessing, even short, sparse spike trains can be converted, so that the standard autoregression modeling can be applied [75], (3) most important of all, spike trains can be filtered to form close approximations to the firing rates or the voltage-fluctuations of the underlying neurons [42].
The rest of this chapter is organized as follows. In Section 2, we briefly introduce the so-called Granger causality index, and then extend it to measure both excitatory and inhibitory effects between network nodes by using the BLP assumption. Section 3 presents three network models to ensure the appropriateness of the proposed algorithm.
In Section 4, we apply the algorithm to analyze real spike train data. Section 5 provides some discussion about the results obtained from Section 3–4, shortcomings of the method, and related future works.