**Ongoing projects.**

- Brain microcircuits and network dynamics of neural traces.
- Non-autonomous dynamics (deterministic and random) in excitable systems. Motivated by cellular electrical excitability and trancription regulation phenomena. Applications on motor neuron responses to common synaptic input and olfactory coding.
- Network topology as determinant of olfactory coding. See past work has been presented at international conferences (Strube-Bloss et al., 2009, 2010)
- Research on human movement. Large-tail distributions, spatio-temporal scaling laws.
- Relationship between ocurrence of dengue infections, distributions of asymptomatic cases, and human movement.

### Recently published research

• *On membrane excitability driven by electrodiffusion*. The model presented and analyzed in this article is an improvement over the classical approach by Hodgkin and Huxley to model membrane excitability (Herrera-Valdez, 2012b). The main idea is to consider the Nernst-Plank equation to derive expressions for ionic transport, and apply the resulting formulations to describe ionic flux across cellular membranes. The transport expressions are used to construct a dynamical system that is similar, but not topologically equivalent to the equivalent system defined with expressions like those used by Hodgkin and Huxley. The model fits experimental data “out of the box” and can be naturally extended to model networks.

• *About the relation between membrane excitability, ion channel expression and bifurcation structure in models of membrane potential* (Herrera-Valdez et al., 2012b). The electrodiffusion model described above (Herrera-Valdez, 2012b) can be used to study the effects of changing patterns of channel expression on the electrophysiological signature of the cell. The relationship between cellular behavior and channel expression can be described explicitly by the bifurcation structure of the model membrane.

• *Regarding the time scales and parallel processing of olfactory stimuli by brain sequentially connected brain regions in the honey bee* (Strube-Bloss et al., 2013). Olfactory stimuli produce a stream of information that reaches first the antennal lobe, and then transferred by projection neurons (PNs) to the mushroom bodies. There, kenyon cells and other neuronal populations further process the olfactory information along with activity triggered by stimuli from other sensory modalities. From there, the extrinsic neurons (ENs, output neurons in the mushroom bodies), carry the processed information to premotor centers. Experiments were done in which simultaneous recordings from PNs and ENs were done. The main, counter intuitive result of this work is that, even after the processing that occurs within the antennal lobe and the mushroom bodies, the population response from ENs occurs before the population response in the PNs. To analyze the data, we constructed a measure to automatically detect and quantify individual neuronal responses.

*On dendritic morphology and signaling.*A graphical method for Together with Sergei Suslov from ASU and José Vega-Guzman currently at Howard University.

**Other published research**

• *On the role played by the historical patterns of local transportation, school closures, and social distancing in explaining and capturing the three waves of AH1N1 influenza observed in Mexico during 2009* (Herrera-Valdez et al., 2011a). The asynchronous epidemic outbreaks that occurred in different states are reproduced by the model. A mechanism of aggregation is proposed to explain how the asynchronous epidemic waves in different states gives rise to the whole-country waves. Importantly, this is the first publication showing that it is possible to produce multiple outbreaks as a combination of different mechanisms not involving artificial sinusoidal forcing.

• *Vaccination models taking into account daily supply and stockpile limitations.* The model used in this paper does not use proportional decay in the vaccinable population (Cruz-Aponte et al., 2011a). The distribution of vaccines according to location and local constraints is simulated using a technique similar to what is used in dynamical programming problems.

• *Reduced models of membrane potential in cardiac myocites* (Herrera-Valdez and Lega, 2010). A 14-dimensional mathematical model by Rasmusson et al. (1995) is analyzed and dimensionally reduced using dynamical systems theory. The resulting 3D and 2D versions of the original model are true reductions with which it is possible to model action potentials of different shapes explained by different patterns of channel expression and the opening probability of channels relative to one another. Importantly, the main currents the membrane potential in the reduced model are the two largest currents from the original model. The analysis and results apply to a large number of mathematical models of cardiac cells found in the literature, leading to the conjecture that excitability in cardiac cells, and in excitable cells in general, is generally described by models in which only two currents, one that tends to increase the membrane potential and one that tends to decrease the membrane potential after an increase.

• *Doctoral dissertation in Physiological Sciences* about the relationship between nearly-coincident spiking and common excitatory synaptic input in motor neurons (Herrera-Valdez, 2009). Dissertation advisor: Andrew J. Fuglevand.

• *Doctoral dissertation in work Mathematics* about the dynamics of minimal biophysical models of excitability (Herrera-Valdez, 2014). Dissertation advisor: Joceline Lega.

### Conferences and contributed talks

Conference presentations about differential contributions of ion channels and electrodiffusion-based transport to membrane excitability (Berger et al., 2009; Cruz-Aponte et al., 2011b; Herrera-Valdez et al., 2009, 2010, 2012a; Smith et al., 2011), extensions of electrodiffusion models to networks (Herrera-Valdez et al., 2011b), the relationship between channel expression and bifurcation structures (McKiernan and Herrera-Valdez, 2012), bifurcation analysis of electrodiffusive membrane dynamics (Herrera-Valdez, 2012c), the role of temperature on electrical signaling (Melendez-Alvarez et al., 2012).