In this section, we illustrate the application of the proposed method to real spike train data and a simulation is also given to examine the validity of the results. Note that significant Granger causality interaction are again calculated using an F -test corrected by FDR for multiple comparison with confidence threshold at P − value = 0.05.

### 3.4.1 Setup and results

Multichannel electrophysiological recording was used for tracking neuronal activity in the anterior cingulate cortex (ACC) and the striatum (STR) and are the same data set

used in our previous study [33]. Briefly, neuronal spikes were recorded from the ACC and the STR of urethane-anesthetized rats after administration of saline or 0.05 or 0.5 mg/kg quinpirole. A multichannel neuronal acquisition processor system (Plexon, Dallas, TX, USA) was used for unit recording, with a filter range of 400 Hz to 8.8 kHz and a sampling rate of 40 kHz. Spikes were further sorted using Offline Sorter (Plexon), based on principle-component clustering with a user-defined template. All animal procedures were approved by the Institutional Animal Care and Use Committee of National Ilan University and adhered to the guidelines established by the Codes for Experimental Use of Animals from the Council of Agriculture, Taiwan.

For this study, data from two independent rats were considered and the numbers of neurons recorded were: 8 (ACC, Rat#1), 9 (STR, Rat#1), 16 (ACC, Rat#2), and 15 (STR, Rat#2). The neurons were randomly put into 2 groups (ACC, Rat#1), 3 groups (STR, Rat#1), 4 groups (ACC, Rat#2), and 3 groups (STR, Rat#2), and each group had 4 (8/2) neurons (ACC, Rat#1), 3 (9/3) neurons (STR, Rat#1), 4 (16/4) neurons (ACC, Rat#2), and 5 (15/3) neurons (STR, Rat#2). After the random grouping described above, the single unit spike trains in each group were pooled as a whole for investigating the brain network. Hence there were 5 (2 + 3) pools in Rat#1 and 7 (4 + 3) pools in Rat#2.

After binning with bin width 2 sec., the GCIs between these random pools can then be computed (Section 3.2.3). Twenty minutes after quinpirole injection, 400 sec. data from both Rat#1 and Rat#2 were used to compute the GCIs. Significant GCIs were found only when certain random group appear, meaning that certain neurons should be pooled together to perform causality. These specific combinations are summarized in Table 3.4, and the corresponding GCIs and NSIs are summarized in Table 3.5. The results show that, under the quinpirole administration, there were excitatory effects inside the ACC (Figure 3.8(a)), excitatory effects from the ACC to the STR (Figure 3.8(b) and 3.8(c)), and inhibitory effects inside the STR (Figure 3.8(d)).

Notice that, for single-input case, e.g., in rat 1 from Pool #2 (ACC) to Pool #3 (STR), both GCI (0.1232) and NSI (0.1116) reflect the degree of causal effect. However, the NSI will be more appropriate than the GCI since the NSI is obtained by fitting a more refined autoregressive model (Step 4 in Section 3.2.3). For multiple-input case, e.g., in rat 2 from Pool #2 (ACC) to Pool #5 (STR) and from Pool #7 (STR) to Pool #5 (STR), the GCI (0.0879 and 0.1598) still reflects the degree of causal effects while the NSI (0.1445 and -0.1058) reflects the degree and the type of synaptic transmission. From this perspective, NSI can be treated as a new complement to provide information on synaptic weights that original GCI does not provide.

Finally, we have to note that, under the saline administration, the same combinations performed no significant GCIs (i.e., GCIs = 0). Furthermore, the GCIs between the ACC of Rat#1 and the STR of Rat#2, and the GCIs between the STR of Rat#1 and the ACC of Rat#2 were all computed, and they were all zero.

Table 3.4: Groups found performing significant NSIs in Section 3.4.1. The numbers of neurons recorded were: 8 (ACC, Rat#1), 9 (STR, Rat#1), 16 (ACC, Rat#2), and 15 (STR, Rat#2). Each group had 8/2 = 4, 9/3 = 3, 16/4 = 4, and 15/3 = 5 neurons.

Hence there were 2 + 3 = 5 pools in Rat#1 and 4 + 3 = 7 pools in Rat#2.

Rat no. Pool no. location elements

1 1 ACC 2,4,5,7

1 2 ACC 1,3,6,8

1 3 STR 3,5,7

1 4 STR 1,4,6

1 5 STR 2,8,9

2 1 ACC 4,8,10,16

2 2 ACC 1,5,6,11

2 3 ACC 3,7,9,12

2 4 ACC 2,13,14,15

2 5 STR 6,7,8,9,15

2 6 STR 1,3,10,11,14

2 7 STR 2,4,5,12,13

Table 3.5: The NSIs between the pooled data from the combinations summarized in Table 3.4.

Rat no. From To GCI NSI

1 Pool #2 (ACC) Pool #1 (ACC) 0.1966 0.1035

1 Pool #2 (ACC) Pool #3 (STR) 0.1232 0.1116

1 Pool #2 (ACC) Pool #4 (STR) 0.1163 0.1995

2 Pool #2 (ACC) Pool #5 (STR) 0.0879 0.1445

2 Pool #2 (ACC) Pool #6 (STR) 0.1098 0.1121

2 Pool #7 (STR) Pool #5 (STR) 0.1598 -0.1058

### 3.4.2 Implications of the pooled data

A spike train obtained by superimposing individual spike trains and disregarding where each spike came from is called a pooled spike train [25]. Adjacent neurons usually work together with each other to generate suitable pooled spike trains to perform specific tasks.

An illustration is provided in Figure 3.9. On the cause side (left brown), the pooled train (pool 1) can be considered as a collective input with respect to the effects of temporal or spatial summation of one of the following two types: (i) the additive effect produced by many PSPs that have been generated from several very close synapses on the same post-synaptic neuron at the same time. (ii) the additive effect produced by many PSPs that have been generated from several synapses which have similar effects on the axon hillock of the same post-synaptic neuron. On the effect side (middle blue), the pooled train (pool 2) can be considered as a collective output, which represents the total discharge of a group of cooperative neurons in function. Again, the collective output (pool 2) can then be treated as a collective input to trigger others (right purple).

Figure 3.8: The firing trajectories (400 sec. with 2 sec. bin) under the quinpirole adminis-tration. (a) excitatory effects inside the ACC (NSI = 0.1035). (b) excitatory effects from the ACC to the STR (NSIs = 0.1116, 0.1995). (c) excitatory effects from the ACC to the STR (NSIs = 0.1445, 0.1121). (d) inhibitory effects inside the STR (NSI = −0.1058). It can be found that positive (negative) NSIs exhibit positive (negative) correlations in the fluctuations of the signals.

Cross correlations can be dramatically amplified by pooling, that is, weak correlations
between pairs of neurons in two populations can lead to strong correlations between the
summed activity of these two populations [56]. Similar results should hold for the GC
analyses. To check this, a simulation is designed as follows: Let P = {p_{1}, p_{2}, . . . , p_{n}}
be a Poisson spike train of time length T with firing rate λ. Let Q be the output spike
train of the almost linear system in Section 3.3.2 with input P , synaptic weight w, and
time delay d. Since P will be treated as a pooled spike train, we uniformly decompose
it into k sub-trains {Pi, i = 1, . . . , k}, that is, each pj ∈ P has the same probability
to be distributed into the sub-train P_{i}, for j = 1, . . . , n and i = 1, . . . , k. As a result,

∪_{i}P_{i} = P , ∩_{i}P_{i} = ∅, and the firing rate of each P_{i} is λ/k. Further, let {U_{i}, i = 1, . . . , m}

be m uncorrelated spike trains with {Pi, i = 1, . . . , k} to serve as the role of environment
neurons. The distribution of each U_{i} is also Poisson with rate λ/k. Now, for T = 10
(sec.), λ = 20 (spikes/sec.), w = 8 (mV), d = 10 (ms.), k = 5, m = 2, and bin width 0.1
(sec.), the GCI from P to Q is 0.2655, a quite large value; while the GCIs from Pi to Q is

Figure 3.9: An illustration for pooled data. On the cause side (left brown), the pooled train (pool 1) can be considered as a collective input with respect to the effects of temporal or spatial summation of one of the two types ((i) and (ii) in the context.) On the effect side (middle blue), the pooled train (pool 2) can be considered as a collective output, which represents the total discharge of a group of cooperative neurons in function. Again, the collective output (pool 2) can then be treated as a collective input to trigger others (right purple).

about 0.005, a very low causality. The results are obtained from the average of 100 such
simulations, and the first 2 sec. of one of the realizations is shown in Figure 3.10; where
P is labeled neuron #1, P_{i}, neuron #2 − 6, U_{i}, neuron #7 − 8, and Q is labeled neuron

#9.

To check the appropriateness of random grouping used in the previous subsection, 5
spike trains are randomly chosen from P_{i} and U_{i} and then be pooled together to compute
the GCI from such pooled data to the target Q. The averaged results are: when these
5 spike trains are all chosen from P_{i} then the GCI attends the maximum 0.2655. When
4 is chosen from P_{i} and 1 from U_{i} the GCI is destroyed and is 0.0641. Finally, when 3
is chosen from P_{i} and the other 2 are from U_{i}, the GCI is 0.0209. The results show that
if the pooled data contains the spikes of other irrelevant neurons then the GCIs will be
small and destroyed.

For real-world spike train data, individual neurons usually perform weak contributions to each other while groups of neurons perform very significant contributions. In the former case, causal influences are difficult to be detected via most statistical methods, grouping and pooling are usually needed to enhance the causation. Since our data set is small, random grouping approach is both reasonable and sufficient to explore the network structure. Significant NSIs are also found within an acceptable period of time. How to efficiently group neurons is absent in our current analysis, but has been being studied via numerical simulations. Efficient grouping strategy is an interesting research topic and

Figure 3.10: The first 2 sec. simulated spike trains of one of the realizations. The
parameters are: T = 10 (sec.), λ = 20 (spikes/sec.), w = 8 (mV), d = 10 (ms), k = 5,
m = 2, and bin width 0.1 (sec.), The source spike train P is labeled neuron #1, the
decomposed trains P_{i} are labeled neuron #2 − 6, the environment trains U_{i} are labeled
neuron #7 − 8, and the target Q is labeled neuron #9.

will be included in a separate article in the future.