This article explores each profession. What is theoretical framework? A theoretical framework is a collection of interrelated concepts, like a theory but not necessarily so well worked-out. A theoretical framework guides your research, determining what things you will measure, and what statistical relationships you will look for.
What is experimental value and theoretical value? The experimental value is your calculated value, and the theoretical value is your known value. A percentage very close to zero means you are very close to your targeted value, which is good. What is an example of theoretical probability? The theoretical probability of an event occurring is an "expected" probability based upon knowledge of the situation. It is the number of favorable outcomes to the number of possible outcomes.
Example: There are 6 possible outcomes when rolling a die: 1, 2, 3, 4, 5, and 6. The only favorable outcome is rolling a 6. How do you find empirical probability? According to several studies, the Kuramoto model shares dynamical similarities with resting-state brain functioning when it shows high metastability [ 14 , 15 ].
This concept refers to high variance of which in other words can be defined as the tendency of a system of oscillators to continuously migrate between a variety of transient synchronous states, allowing a dynamical organization between the elements of the network. The system continuously goes from ordered to disordered states [ 16 ].
Then, the values of the parameters in the model were selected so that the global dynamics showed high metastability. Note that any change in can be considered a change in the mean velocity of the conduction delays between oscillators, and other values of with a different range of produced similar behaviors.
For example, we found approximately the same effects for 2. The Kuramoto model was simulated for a wide range of values. As indicated before, represents the strength in the global connectivity of the model, and from a biological point of view it could be seen as a parameter to characterize integration between oscillators. As in Cabral et al. It would be important to state that since no experimental data are provided here, our results are obtained for parameters of the model , , and metastability that have been shown, in previous works [ 18 — 20 ], to parallel key characteristics of brain functioning.
Hence, excluding the parameter , we did not manipulate the parameters in the coupled equations of the Kuramoto system.
Integrated information was measured with the version and it is mainly based on the concept of effective information [ 9 ].
Let be a multivariate random variable that takes values in the space. It is evident that the dimensions of are the number of elements in the system that generates. The effective information generated by a system in its current state about the state with respect to a bipartition of it is defined by the mutual information generated by the entire system minus the sum of the mutual information of its parts in the bipartition:.
Mutual information in bits can be calculated with the expression a measure that gives the average bits that can be predicted in given the state [ 21 ]. The calculation of mutual information includes the calculation of probabilities and joint probabilities of any estate and.
To apply to the Kuramoto model described in the previous section, we divided the 66 regions of the original network into 6 clusters proposed by Hagmann et al. In addition, and following [ 22 ], time series for each cluster were calculated as the synchrony between the oscillators belonging to that cluster.
Then, we characterized these series as synchronized or not synchronized by constructing new binary time series from each. We selected this value because it was the median of , and using the median for thresholding eliminates the possible influence of extreme values, due to its robust properties. In addition, there was a theoretical reason for this election.
Finally, was calculated to these binary series and the result was taken as the complexity value of the Kuramoto model. In order to calculate an estimation of the PCI in the Kuramoto model, it was necessary to solve two problems in the simulation process. The first one was to perturb or stimulate the system from an external source to emulate the effects of TMS , and the second difficulty we found was to calculate reliable ERPs for the final PCI calculation.
The problem of the stimulation was easily solved since it has been done in other studies. For example, Hellyer et al. Hence, in our study we followed these studies and perturbed the system by transient increases in the connectivity between six oscillators located in the parietal cortex see Figure 1. We randomly repeated this stimulation 15 times for each numerical integration of the model. The next step was to build reliable ERPs with the resulting phases from the oscillators.
To achieve this goal we simulated EEG series, and then, ERPs were calculated for each model by averaging the segments associated with each period of stimulation. We explain this procedure in detail in the next two sections. The EEG activity from 32 sensors was simulated for each condition in agreement with the following weighted sum of the activity in each oscillator: where is the time series from sensor th and is the weighted contribution of source th in sensor th.
Each was calculated using a standard forward model algorithm [ 25 ] applied according to the Talairach coordinates of the oscillators. After that, each oscillator was considered a cortical source. Second, the weights of these sources were normalized to a maximum value of 1. Because in the ERPs the interesting information is in the amplitude, we calculated the envelope of with the Hilbert transform. Envelopes of the signal were then used to construct the ERPs with the average of all segments in each realization of the model.
Formally, ERPs were built with the analytical signal in the complex plane of which is where is the original signal and represents the real part of the new complex series and is the imaginary part from the Hilbert transform with as the imaginary operator.
The modulus is the amplitude or the analytic power of the signal and can be easily calculated with. These new series were considered the activity from each sensor and the ERPs were built with segments extracted from them. PCI was obtained following the original algorithm in Casali et al. The signals were downsampled ten times to obtain a sampling rate similar to real data. SS was used as input for the Lempel-Ziv measure [ 26 ] to estimate the algorithmic complexity.
The algorithm seeks for the minimal number of patterns necessary to describe the sequence. PCI is defined as the normalized value of :. The results for the Kuramoto simulations are characterized in the first place to understand the basic dynamics of the model.
In addition, we include graphical descriptions of the ERPs to visualize the structure of the averaged waves at each sensor from simulated perturbations. In Figure 3 a we show a diagram with the behavior in the baseline condition at several values of its coupling parameter. The most important property in the evolution of is the metastability that can be estimated by the variability of.
In addition, it is important to note that the frequency structure in is not fixed for all s. One can observe in Figure 3 a that the frequency of seems to increase with the increase of. To better understand this phenomenon, we include a spectral decomposition of the evolution of in Figure 3 b. Surprisingly, we found a complex landscape in the oscillatory structure of. The general structure of the spectral diagram showed a resemblance with the bifurcation diagram of the classical logistic map.
This similarity appeared because it showed a bifurcation-like proliferation of frequency components as the parameter increased. The nature of this spectral structure, however, was not explored and goes beyond the goals of this study. If we inspect Figure 3 b , it can be stated that the end of slow oscillatory properties of high metastability was evident at. As increases further, larger and larger synchronized clusters are formed, resulting in a reduced number of components, until approaches 1, with ultimately only a single component as tens to infinity [ 27 ].
Accordingly, in Figure 3 b , the main frequency of the signal slowly increased with , and the components of seemed to increase with as well following a complicated pattern.
It is also noteworthy that in Figure 3 b another bifurcation-like region for can be perceived that consists in a reduction in the number of components. Hence, by the end of the landscape, seems to be more simpler with less oscillatory properties and probably this could lead to low values of and PCI.
The shape of this diagrams led us to consider that and PCI could be sensitive to the bifurcation region in. If metastability is a necessary condition for brain functioning [ 16 ] it would be reasonable to think that should diminish in the low metastability region. The same should be true for PCI if this measure is closely related to. As we will show in the next sections, the reduction after metastability was only found for. The relation between and PCI was assessed using Pearson product-moment coefficient between the values of obtained for each of the -levels from.
Thus, apparently, and PCI are linearly independent. A graphical description of the evolution of and PCI and the metastability of the model can be observed in Figure 4. Values were calculated with. However, the PCI did not show a significant decrease in this region of.
In fact, the PCI evolution seems to progressively increase with. The fact that there is no decrease in PCI does not mean that this measure is not related to ; as seen before, a closer exploration of both measures indicated a positive relation between them.
Under the assumption that conscious states come from integrated information in a system, various metrics have been proposed to try to quantify consciousness.
In the present work, we tested two of them using a neurocomputational model. On the other hand, the PCI distinguishes conscious versus unconscious states at a single patient precision [ 10 ]. Under the assumption that conscious states correspond to a distributed but nonuniform spatiotemporal pattern of current sources, Casali et al. Despite the excellent results at applied level, the claim that the measure is theoretically grounded in a conceptual understanding of consciousness deserves a closer look.
In the present work, we have tackled the possible relationship between these two measures of the degree of consciousness in a system. As stated previously, according to the IIT, wakeful consciousness requires the ability to integrate information across multiple brain regions with a high degree of differentiated activity. Thus, loss of consciousness may result as a consequence of a loss of integration as well as a loss of differentiation or both.
Still Confused? Nope, got it. Play next lesson. Try reviewing these fundamentals first Introduction to probability Organizing outcomes Probability of independent events. That's the last lesson Go to next topic. Still don't get it? Review these basic concepts… Introduction to probability Organizing outcomes Probability of independent events Nope, I got it. Play next lesson or Practice this topic.
Play next lesson Practice this topic. Start now and get better math marks! Intro Lesson. Lesson: 1. Lesson: 2a. Lesson: 2b. Lesson: 2c. Lesson: 2d. Intro Learn Practice. What is experimental probability? What is theoretical probability For theoretical probability, it doesn't require you to actually do the experiment and then look at the results. Theoretical vs experimental Why is there a difference in theoretical and experimental probability? Practice problems We'll now see how experimental and theoretical probability works with these questions.
The results are in the chart below: What is the experimental probability of both coins landing on heads? Solution: We are looking for the experimental probability of both coins landing on heads. Solution: Now, we are looking for the theoretical probability. Question 1d: What can we do to reduce the difference between the experimental probability and theoretical probability? Do better in math today Get Started Now.
Introduction to probability 2. Organizing outcomes 3.
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