**Historical patterns.** To extract information on the changes in the emotional state of humans, we used a searchable COHA database to determine the frequency of usage of about 20% of words from the ANEW database by [8] (115 words). Their frequency of usage was determined as a function of time. Figure 3 shows two examples: the word “tired” increased in frequency and the word “happy” decreased. Linear regression was used to calculate the slope of increase/decrease of all the words’ usage from 1810—2000.

Then the frequency slopes of all the 115 words were plotted vs the words’ energy and valance, to assess which kinds of words (high/low in valence or high/low in energy) increased in frequency over time, and which ones had decreased in frequency over time. Figure 4 (left) shows the slope vs energy graph for these words. It turned out that high energy words had more negative slopes, indicating a decrease in usage, while the low energy words had more positive slopes (increased in usage). Linear regression was used to find the pattern, and the dependence was statistically significant (p = 0.0049).

Further, slope vs. valence was plotted (see Figure 4, right), and while linear regression did not reveal a significant pattern, quadratic fitting was used. It is seen by the parabola’s curve that words with both high and low valence decreased in usage, while words that were neither positive nor negative (the section in the middle of the graph), increased. The p-value for a quadratic fit for this graph was <10-6. Both results in Figure 4 are statistically significant and demonstrate an increase in more indifferent words over time, rather than passionate and energetic ones.

Figure 2 shows the valence-energy coordinates of the subset of words used in the analysis. One immediately notices the parabola-like shape that appears. Comparing this with valence-energy values of all the words in the ANEW database [8] one can see that our subset faithfully represents the trend common to all the words in the dataset, namely, words that are characterized with very low or very high valence values tend to be more energetic, and words with neutral valence are typically low on energy. The fact that valence and energy are not entirely independent is also consistent with the result of figure 4: energetic words, as well as words with very high and very low valence, experienced a decline, while “neutral words” (that is, words with intermediate valence values) as well as words with low energy values increased in frequency over the years.

**Contemporary patterns.** Next, we turn to contemporary text analysis. 180 blogs from Kaggle were analyzed. These were divided into groups by age (young, intermediate, older), gender (male or female), and career (arts or sciences). The blogs were analyzed by using an online sentiment analysis tool, https://text2data.com/, and their energy and valence values were determined. Then inter-group comparisons were performed by a Kolmogorov-Smirnov test.

Figure 5 shows histograms for the energy values of the blogs split into groups. On the left the blogs are split by age, with the younger group on top and the older on the bottom. The mean values for each group are indicated, and the result of the Kolmogorov-Smirnov test is shown next to the vertical arrow. Similarly, the middle graphs split the blogs by gender, and the graphs on the right plot them by occupation. One can see that all three p-values are larger than 0.05 indicating that there is no significant difference between the groups. Thus, the different ages, careers, and gender of people do not have an effect on the blog’s energy levels.

Although there were no statistically significant differences in the energy levels between groups, there did prove to be significant differences between the valence levels of different groups, see Fig. 6. The valence of blogs from different age groups were compared (graphs on the left). The blogs from a younger/middle age group (13–30 years old), had average valence − 0.24, while the blogs from the older group had valence − 0.04 (p = 0.019). The lower valence value for the younger group indicates that the younger generation displays significantly more negative emotions.

Next, the mean valence of females’ blogs was determined to be -0.26, while for males’ blogs it was -0.10 which is a higher number than the female average (see the graphs in the center of Figure 6). Thus, it is shown that females reflect more negative feelings through their writing (p=0.034 by Kolmogorov-Smirnov test).

Finally, blogs were chosen from two different segments of society, based on occupation: science and art. When their computed values of valence and energy were graphed and compared (Fig. 6 on the right), it was shown that people with an artistic background showed more negative emotions. The mean valence for science bloggers was 0.13 and for art bloggers − 0.16, with the p value being 0.014. The average is significantly higher for the science blogs, meaning that the people whose career is in the arts express more negative emotions.

**Mathematical modeling.** Mathematical modeling can be used to explore the trends evident from data analysis, and identify mechanisms that are compatible with the dynamics that are observed. We will use imitation dynamics to describe the behavior of different emotions that are expressed in written texts. To begin, we need to determine the appropriate space that best describes the emotional coordinates. Even though two- and even three-dimensional spaces are often used (such as the valence-energy space, or valence-arousal-dominance), for our purposes this can be simplified.

This lack of independence of the two emotional coordinates (see Fig. 2) allows for reduction of the dimensionality of the emotion space, by placing all words along a single axis, \(z\in R\), where the absolute value, *|z|*, is related to the energy of the word and the sign of *z* corresponds to positive/negative valence. Denote by *x(z,t)* the frequency at which emotion with coordinate *z* is represented in texts at time *t.* To describe the change of this quantity, we need to associate the emotional coordinate with a “fitness” parameter, *r(z)*. This parameter characterizes the amount of appeal each emotion has with the public/consumers of the written media. Here we will assume that the emotion with coordinate *z* “resonates” with those who experience a similar emotional state, and the closer *z* is to the emotional state of the consumer, the higher the appeal. This gives rise to the model with \(r\left(z\right)={\int }_{-\infty }^{\infty }\rho \left(y\right)S\left(\left|y-z\right|\right)dy,\) where \(\rho \left(z\right)\) denotes the probability distribution of the consumers to experience emotion with value *z*, and *S(|y-z|)* is a function that measures similarity between emotions with coordinates *y* and *z*. In the Materials and Methods Section we showed that the fitness function, *r(z)*, experiences a maximum at *z = 0*, under a range of relevant assumptions, such as that the probability distribution function on the space of emotions is a bell-shaped curve.

To implement the dynamics described by Eq. (1), we discretize the space of emotions, such that \({x}_{i}\left(t\right)\) describes the frequency of emotion *i* at time *t*, where *i* is an integer between *-L* and *L*. Eq. (1) models the process whereby emotion *i*’s representation at a future moment of time is dictated by (1) how much creators of the media have been exposed to *i* in the present and (2) how much appeal this emotion has among the consumers. Figure 7 illustrates these calculations, where all emotions are characterized by an index *i* is an integer between − 20 and 20. Panel (a) shows an example of a fitness function construction, where we assumed that the probability distribution of emotion values is given y \({\rho }_{i}={Ce}^{-\frac{{i}^{2}}{150}}\), where \(C\approx 0.0469\) is a normalization constant; the similarity function is \(\text{S}\left(\text{q}\right)=\frac{{e}^{-\frac{{q}^{2}}{20}}}{\sqrt{20\pi }}\), and \({r}_{i}=\sum _{k=-L}^{L}{\rho }_{k}S\left(\left|i-k\right|\right)\). Eq. (1) was solved under a random initial condition, and a typical result is shown in panels (b-d). The individual trajectories, \({x}_{i}\left(t\right)\), are plotted as functions of time in panel (b). We can see that while some trajectories decrease, others increase. These trajectories are related to (but are not the same) as time-series of individual word frequencies such as those shown in Fig. 3. The difference is that instead of tracing individual word frequencies, we model the frequencies of emotional coordinate values that words represent.

Panel (c) shows how the frequencies of different evolutional values change over time, by showing 5 different time-points. Here, the color blue represents time zero (the random initial condition, see also the inset), and then the colors yellow, green, red, and purple show the frequencies at times t = 50,100,150, and 200, respectively. We observe that the frequencies of emotions near the middle of the range increase, while the frequencies corresponding to strongly negative and strongly positive emotions decrease. Finally, panel (d) demonstrates this trend by plotting the time-derivative (that is, the slope) of the trajectories, evaluated at t = 40. This graph can be compared with Fig. 4(b), which shows a similar, parabola-like shape.

The above results show that an increase in frequency of middle-range emotions is a natural consequence of a bell-shaped curve of consumer emotional states. In order to visualize the dynamics of emotions in texts that results from a change in consumer emotional state, we performed the following simple calculation (figure 8). We started with a consumer distribution that has a certain preference for positive emotions, see figure 8(a), and proceeded to generate the corresponding spectrum of emotions in the media, see panel (b). Then, we assumed that the emotional state of the consumers experienced a change, such that it is now skewed towards a preference for more negative emotions, see panel (c). Panel (d) shows the change experienced by the frequencies of different emotions as a result of that change. In particular, the colors blue, yellow, green, red, and purple represent time points t=0,70,140,210, and 280 after the change took place. We observe a gradual shift in the distribution of emotional value, with the graph of the slopes (panel (e)) showing a decrease for positive valence values and an increase for more negative ones.