# 4.08 Polynomial Identities And Proofs Assignment

### Formation of functional topology

To determine whether A1, S1, and V1 functional circuit wiring diagrams exhibited invariant features, we monitored neuronal activity in 43 slices from each region of the mouse neocortex (11 of A1, 21 of S1, and 11 of V1) using high speed multi-photon calcium imaging [15], [25], [31]. Spontaneous circuit activity requires intact excitatory amino acid transmission [15], [33], sufficient oxygenation [34] and corresponds to UP states within single neurons which comprise the functional circuit [15], [35]. Previous reports have found that spontaneous activity delineates all of the possible multi-neuronal patterns within a sampled population and that a sensory input activates only a subset of these patterns [14], [36]. By monitoring spontaneous activity in the imaged field of view, we hoped to maximize the number of pairwise correlations within the imaged populations. We imaged the flow of activity through large populations of neurons (A1: 595±101 cells, S1: 704±157 cells, V1: 734±129 cells) at the mesoscale in a two-dimensional circular imaging plane with a diameter of 1.1 mm that comprised multiple layers and columns with single-cell resolution (Figure 1A). We confirmed activity was not biased to any one lamina and that our sampling was uniform across our field of view, since the amount of activity observed across all circuit events did not differ between layers (, KW-test; see *Methods* for explanation of laminar identification). Because temporal resolution of multi-photon microscopy is compromised at these spatial scales, we used the heuristically optimized path scan technique [25] (Figure 1B), which allowed us to achieve fast frame rates (frame duration 86±17.7 ms) that did not differ between regions (, KW-test). We deconvolved calcium fluorescence changes of each detected neuron into spike trains (Figure 1C) [31] and generated rasters of spiking activity for the entire imaged population of neurons (Figure 1D). All regions of the sensory neocortex showed a common capacity for emergent, multi-neuronal patterned activity, characterized by discrete periods (>500 ms) of correlated action potential generation within subsets of neurons. Circuit events were separated by periods of quiescence and we refer to these distinct, clustered epochs of spontaneous action potentials as individual circuit events. The start and finish of a circuit event was easily resolvable because the field of view was either quiescent, corresponding to a DOWN state in a single neuron, or was active, corresponding to a UP state in a single neuron [35]. One circuit event lasted 1203±456 ms in A1,1568±885 ms in S1, and 1342±698 ms in V1. We imaged 82 total circuit events in A1, 268 total events in S1, and 104 total events in V1.

Figure 1

**Sensory cortex exhibits spontaneous circuit activity.**

Using this data, we generated graphical abstractions, or circuit topologies, corresponding to functional activity over all circuit events observed in a single field of view. Neurons were represented as nodes in each graph. Edges between nodes were directional and formed according to the following rule: neuron A was considered functionally connected to neuron B if neuron B fired in the subsequent frame (Figure 1E). These edges were then weighted according to how many times this single frame lagged correlation occurred, normalized to the number of events in that field of view. Thus, stronger edge weights indicated reliable, correlated spiking, whereas weaker edge weights indicated unreliable, weakly correlated spiking (Figure 1E). The resultant graphs contained a large number of edges (median: 3.4×10^{4} functional connections, range: 4.2×10^{5} functional connections). Note that although a functional relationship between neurons increases the probability of them having a synaptic connection [18], [33], a linear relationship between each functional edge and a synaptic connection does not exist [16]. Rather, given our method of inference, the functional connectivity measure captured the flow of activity through the network during a circuit event.

### Neocortical functional circuits are characterized by invariant features

#### Functional circuitry is composed of non-random and occasionally distal connections

We found that most functional connections were locally organized and biased toward shorter pairwise distances, consistent with previous functional and anatomical studies [5], [15] (Figure 2, right column). To gain insight into the spatial dependency of functional circuit wiring, we compared functional topologies generated from the data with null models of varying spatial constraint. To this end, we generated a matched random and -nearest neighbors null graph for each functional topology. The random and -nearest neighbors topologies represented upper and lower bounds of spatial constraint, respectively. In each random topology, nodes were placed in the same locations as a corresponding functional topology, but each node had a fixed probability of functional connection () with any other node. We found that random topologies were spatially relaxed because their connections were not constrained to subsets or neighborhoods of nodes. Importantly, the spatial distribution of functional connections in random topologies was statistically indistinguishable from the long-tailed probability distribution of pairwise distances in the field of view (; KS-test; Figure 2, left column). Thus, the random topologies still contained a distance dependence in its likelihood of a connection. The -nearest neighbors topology was a null model consistent with previous anatomical studies that described synaptic connectivity in a nearest-neighbors paradigm [5]. In each -nearest neighbors topology, nodes were placed in the same locations as the corresponding functional topology, but neuron A was functionally connected to neuron B if and only if B was one of -nearest neighbors of A (Figure 2, middle column). In this case, the probability of functional connection was heavily biased towards local neighborhoods, and thus the connections were spatially restricted. Because connections in random and -nearest neighbors topologies are non-specific beyond their spatial constraints, the poor quality of their fit to the data also provides insight into the prevalence of non-random functional connectivity that is not simply dependent on short distances.

Figure 2

**Functional topologies are composed of non-random, proximal and distal connectivity.**

Let denote a functional connection derived from the data and denote a functional connection in one of the corresponding nulls. To explore how well random and -nearest neighbors topologies explained the data, we computed the following conditional probability distribution with respect to each model:

We expected the above expression to evaluate to 1 if there was a one-to-one relationship between edges in the -nearest neighbors or random topology and edges in the functional topology created from the data.

To calculate the above expression, we employed Bayes' rule:

The -nearest neighbors topologies captured 12±9 percent of the total number of functional connections in the data, while the random topologies captured 9±8 percent of the total number of functional connections in the data. The amount of the data captured by the null models did not differ between regions (-nearest neighbors: , KW-test; random topologies: , KW-test). Thus, functional topology is non-random, and connections that extend beyond local neighborhoods form a substantial portion of connections at the mesoscale.

To further characterize the distributed nature of functional topology, we next analyzed functional connections traveling between and within the lamina visible in our field of view (L1, L2/3, L4, and L5; see *Methods* for explanation of laminar identification). Due to relaxed spatial constraints and non-specific connectivity, Random topologies contained significantly more functional connections traveling between layers than within layers (between: 52±11 percent, within: 33±11 percent, ). The difference in the number of functional connections traveling between and within lamina in random topologies was significant across areas (, , ). In contrast, -nearest neighbors topologies had significantly more functional connections traveling within layers than between layers (between: 11±3 percent, within: 82±11 percent, ). The difference in the number of functional connections traveling between and within lamina in -nearest neighbors topologies was significant across areas (, , ). Functional topologies generated from the data had no significant difference between the number of functional connections traveling between layers and the number of those traveling within layers (between: 46±12 percent, within: 46±14 percent, p = 0.93, KW-test). Furthermore, the difference in the number of functional connections traveling between and within layers was insignificant across areas (, , ).These analyses suggest that neocortical functional topologies consist of non-random, occasionally distal connections that, despite being skewed in probability toward local neighborhoods, are not solely governed by spatial proximity and are distributed across the field of view.

#### Neuronal influence in local functional circuitry is log-normally distributed

Neuronal networks have been found to contain neurons which are connected to large numbers of other cells, called hubs [15], [21], [37], [38]. Traditional approaches characterized a hub as having a large degree that is multiple standard deviations from a network's norm [39]. However, this metric of degree centrality fails to fully capture the influence of a node in a network. To identify network hubs that focused on functional information flow, we utilized the eigenvector centrality measure of node influence. Let denote an × adjacency matrix. Then the eigenvector centrality of node is defined as the entry in the normalized eigenvector corresponding to the largest eigenvalue of .

The above implies is a linear combination of centrality scores of all nodes connected to ; a node that has a high eigenvector score is connected to nodes that are also high scorers. Uniqueness of the eigenvector associated with the largest eigenvalue is ensured by the *Perron-Frobenius Theorem*, which states that any positive definite square matrix has a unique largest real eigenvector with strictly positive components [40]. The difference between eigenvector centrality and degree measures is revealed in the following example. Let one neuron project an edge to another neuron, which in turn projects to ten neurons. The first neuron in this chain would be assigned a degree of 1, and thus would be considered an insignificant actor in the circuit under degree centrality. However, under eigenvector centrality, each neuron's score is a linear combination of all other neurons' scores. Eigenvector centrality would assign the first neuron a high score as it is considered to be the most influential driver of local activity (Figure 3A). Projections of both measures onto an imaged field of view qualitatively revealed differences in the contour distributions of assigned scores by degree and eigenvector centrality (Figure 3B). We calculated the distribution of eigenvector centrality and degree scores, and found that the former fit to a log-normal distribution in all three areas of the sensory neocortex ( = −3.88±0.07, = 1.01±0.05, = 0.13; = −4.01±0.07, = 1.05±0.05, = 0.08; = −4.08±0.08, = 1.21±0.06, = 0.13; KS-test; Figure 3C), and that the latter fit to a normal distribution in all three areas of the sensory neocortex ( = 0.05±0.002, = 0.03±0.001, ; = 0.04±0.002, = 0.03±0.002, ; = 0.05±0.002, = 0.03±0.002, ; KS-test; Figure 3C). Interestingly, we found that eigenvector centrality scores were not correlated with in-degree ( = 0.14±0.24, ; Pearson correlation; Figure 3D), but were highly correlated with out-degree ( = 0.98±0.04, ; Pearson correlation; Figure 3D). The strength of the correlation did not differ between regions (; KW-Test). The tight relationship between eigenvector centrality and out degree implies that the influence of neuron in its local circuit is highly related to its feedfowardness. Thus, it is interesting that V1's eigenvector centrality distribution is translated to greater eigenvector centrality scores relative to A1 and S1, given V1's higher propensity for feedforward activity (Figure 3C) [41], [42].

Figure 3

**Hub neurons defined with eigenvector centrality.**

#### Functional circuit topologies are connected

A graph is connected if there exists a sequence of edges from any node to any other node. To quantify the connectedness of neocortical functional topologies, we employed the following theorem:

**Theorem:** Let the undirected graph be specified by an adjacency matrix and have a degree matrix , where **1** is the column vector of all 1 s. Let the Laplacian have eigenvalues . is connected if and only if . denotes the algebraic connectivity of . (For proof, see [30])

The larger the algebraic connectivity is, the more strongly connected the graph is. An algebraic connectivity close to zero indicates a graph that is highly modular and susceptible to attack, which makes connectedness a prime topological metric for defining robust networks [43]–[45]. We assessed the connectedness of the functional topology by first transforming all directed edges to undirected ones, as this is required for the theorem to be applicable. We therefore lost information on circuit flow provided by directed edges, but preserved information on the abstract structural features of the topology, like the general reachability a neuron in the circuit. We then computed the second smallest eigenvalue of the Laplacian of the resulting adjacency matrix, normalized by the number of nodes in the graph. We found that functional topologies in each sensory area were connected ( = 0.385±0.17; Figure 4A) and that the amount of connectivity did not differ between regions (; KW-test). These values significantly differed from the moderately connected random topologies ( = 0.24±0.006; , , ; KW-test) and the weakly connected -nearest neighbors topologies ( = 0.02±0.004; , , ; KW-test). This analysis suggests that an arbitrary path from any neuron to every other neuron is present in functional circuit topologies. Interestingly, the variance of the algebraic connectivities of the models was much smaller than those of the data. The greater variance of algebraic connectivity present in the data might emerge from specific patterns of functional connectivity that are absent in the non-specific random and -nearest neighbors topologies.

Figure 4

**Functional topologies are connected and their size is independent of the number of neurons sampled.**

#### The size of functional circuit topologies does not scale with the number of neurons in the field of view

Sequences of neuronal activations in the neocortex likely represents a neural syntax that encodes external stimuli [46]. Since each directed edge in a functional topology represents a sequential activation of two neurons, each activation sequence can be defined as a walk, or a sequence of visitations to adjacent nodes, in the functional topology. Because spontaneous activations delineate all possible multi-neuronal patterns within a sampled population [14], [36], we quantified the number of possible activation sequences of a given length in functional topologies generated from spontaneous activity. To compute this metric, we employed the following theorem:

**Theorem:** Let the graph be specified by an adjacency matrix . For any , the entry of the matrix is equal to the number of walks from to in of path length .

This theorem can be proved through induction; we use the facts that each edge in the graph is unique, and that to form a walk of length from vertex to , one must first have a walk of length from vertex to , and then a walk of length 1 from vertex to . Note that the number of open sequences, walks that do not have equal starting and ending nodes, is the sum of the upper and lower triangular matrices of (Figure 4B). In addition, the number of closed sequences, walks that have equal starting and ending nodes, is the trace of (Figure 4B). Open sequences may relate to feedforward activity, while closed sequences may relate to recurrent activity [15], [42]. In all analyses, we computed the number of open sequences of path lengths 1 to 10 and the number of closed sequences of path lengths 2 to 10, since a closed sequence of length 1 does not exist. We found that the number of possible open sequences and closed sequences, as a function of path length, were perfectly fit by exponential functions across all functional circuits ( = 1.00±0.00). This exponential growth reflected the combinatorial explosion of possible sequences of larger lengths, as the graphs analyzed contained a large number of nodes and edges.

Next, we computed the ratio of the number of open sequences to the number of closed sequences in each graph, excluding sequences of length 1. We refer to this ratio as the *O-C ratio*. We found that across all path lengths analyzed, there were 217±47 open sequences for every closed sequence in A1, 293±151 open sequences for every closed sequence in S1, and 339±67 open sequences for every closed sequence in V1. The higher O-C ratio in V1 likely supports the postulate that the region has a greater propensity towards feedforwardness [37], [42].

The O-C ratio did not differ between path lengths (, , ; KW-test), suggesting that while the *raw number* of open and closed sequences grows exponentially as a function of path length, the *ratio* of open to closed sequences stays constant. In contrast, the O-C ratio in random topologies increased 2-fold from length 2 to length 3 (, , ; KW-test), and stayed constant for larger path lengths (, , ; KW-test). Further analysis showed that random topologies had a far greater percent of possible reciprocal connections (closed sequences of length 2) than functional topologies generated from the data (random: 24.9±0.04 percent; data: 8.4±8.2 percent).The greater prevalence of reciprocal connections in random topologies likely results in the smaller O-C ratio at length 2.

Because the number of open and closed sequences as a function of path length grew exponentially, we could linearize the curves by transforming them into log-scale (Figure 4C). Linearization allowed us to use slope as a feature of how the number of sequences varied with path length. We found that the distribution of slopes did not differ between open and closed sequence growth curves for all functional topologies generated from the data (; KW-test). This finding confirmed the invariance of the O-C ratio to path length in the data.

We found that the slope of a sequence growth curve was strongly correlated with the number of functional connections in the corresponding topology (Open and closed sequences: , ; , ; , ; Pearson correlation). This finding prompted us to characterize how the number of sequences in a functional topology varied with the number of neurons in the field of view. We hypothesized that random topologies were greedy: the more nodes in the field of view, the more activation sequences would be possible, because every node in the random topology has a 0.5 probability of being connected to any other node. Thus, the size of the random topology would scale with the number of nodes in the field of view. Supporting this hypothesis, we found that the slope of sequence growth curves for random graphs were strongly correlated with the number of nodes in the corresponding random graph in all regions (Open and closed sequences: , ; , ; , ; Pearsons linear correlation). In contrast, we found that the slopes of sequence growth curves in the data were uncorrelated with the number of neurons in the corresponding functional topologies in all regions (Open and closed sequences: , ; , ; , ; Pearson correlation; Figure 4D). This finding suggests that the size of functional connectivity does not scale with the number of neurons in the field of view, and that only a subset of neurons in the field of view are recruited during any one circuit event. These analyses further support the postulate of specificity in functional connectivity, and suggest that the lack of strong positive correlation between the slope of the sequence growth curve and number of neurons in the field of view is inherent to the functional connectivity patterns of these regions.

### Local circuit flow covers entire angular space

There is an ongoing debate on whether the cortical column, which is oriented perpendicular to pia, regulates and shapes the flow of information in sensory cortices [47]. Coronal slices allowed us to image activity patterns with near simultaneity across all lamina. Using this data, we assessed directional flow in functional graphs by computing the angle and distance between the source and destination of directed functional connections relative to the orientation of pia. *Flow maps* are plots that capture direction of circuit flow with points scattered at a radius and angle about the origin. represents the distance of the functional connection from the source to the sink, and represents the angle between the source and the sink.

We measured the amount of angular clustering of activity flow in sensory areas by computing the circular variance of functional connections. The clustering of points at a particular angle indicates stereotypy of functional flow across events in a neighborhood of the functional topology. We calculated the amount of angular clustering by computing the circular variance of the set of points.

Circular variance is defined as:

The value of the circular variance varies from 0 to 1; the lower the value, the tighter the clustering of points about a single mean angle. In functional circuit topologies from all three areas of the sensory neocortex, flow covered the entire angular space, regardless of the pairwise distance, or radius, spanned by the functional connection (Figure 5A). We found that the spread of circular variance increased for functional connections which spanned the largest distances, most likely due boundaries imposed by pia, internal capsule, or field of view (Figure 5B). Thus we did not find a canonical circuit flow in spontaneous cortical activity regardless of sensory area.

Figure 5

**Circuit activity flow uniformly covers angular space.**

### Large fields of view are necessary to investigate functional topologies

## Polynomial Identities

Polynomial Identities and Proofs

Essential Questions:

• How can polynomial identities be proven?

• What can polynomial identities apply to beyond just polynomials?

It's time to show off your creativity and marketing skills!

You are going to design an advertisement for a new polynomial identity that you are going to invent. Your goal for this activity is to demonstrate the proof of your polynomial identity through an algebraic proof and a numerical proof in an engaging way.

You may do this by making a flier, a newspaper or magazine advertisement, making an infomercial video or audio recording, or designing a visual presentation for investors through a flowchart or PowerPoint or even word document.

You must:

• Label and display your new polynomial identity

• Prove that it is true through an algebraic proof, identifying each step

• Demonstrate that your polynomial identity works on numerical relationships

Create your own using the columns below. See what happens when different binomials or trinomials are combined. Square one factor from column A and add it to one factor from column B to develop your own identity.

Column A Column B

(x − y) (x2 + 2xy + y2)

(x + y) (x2 − 2xy + y2)

(y + x) (ax + b)

(y − x) (cy + d)

#### Attachments

#### Solution Preview

• How can polynomial identities be proven?

Polynomial identities can be proven by performing operations such as FOIL (First Outside Inside Last), multiplying using the Box method, or by substituting numbers into the variables.

When proving an identity, the goal is to transform one side of the identity so that it is identical to the other side of the identity. The work shown should not transform both sides of the identity simultaneously. Instead, we start with one side of the identity (such as the left side) and use known operations such as addition, subtraction, multiplication, division, or FOIL to prove that the side we started with (left side) is equal to the other side (right side) of the identity.

Note that an ...

#### Solution Summary

A Word file containing the solution is provided. Answered in 542 words.

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