Extracting information from noisy environments is an ongoing challenge for biological systems. When listening to music or podcasts on our personal devices, we can efficiently tune out background noise with noise-cancelling headphones, thanks to modern technology. However, our nervous system is constantly bombarded by the cacophony of external and internal surroundings. Our recent work set out to examine the type of cognition that is built into neurons for signal detection and noise modulation.
We focused on sensory neurons, which are tasked with encoding meaningful signals in noisy environments. Here I show how mathematics, physics, and biology work together to engineer the neurocognition of signal detection and noise modulation.
We begin with mathematics, and briefly describe the structure of a dynamical system for signal encoding in the nervous system. Neurons encode stimulus information by generating a dynamic code that consists of electrical impulses known as action potentials or spikes. The timing between spikes, known as interspike intervals (ISIs), contain stimulus information. This information must remain robust and immune to spurious inputs or noise. Neurons can ensure a robust code by grouping spikes into bursts, like words in a sentence. Bursting consists of groups of spikes with short ISIs that are separated by wider gaps called interburst intervals (IBIs). Describing a dynamical system that encodes bursts (a sequence of words) technically requires a system of differential equations with a minimum of two subsystems: a fast subsystem (which consists of variables that can change quickly, possibly during spike generation), and a relatively slow subsystem or variable. The slow variable’s up and down swings can cause the qualitative switch between bursts (words) and quiescence (gaps). This occurrence is often explained by the system’s geometry (phase plane) and bifurcation of the fast spike-generating subsystem by the slow subsystem/variable. This is the basic dynamic structure of a word/burst encoding machine.
Next, we turn to physics. Imagine that the aforementioned burst encoder, which is essentially an oscillator, is subject to inputs that consist of multiple frequencies. Think of a tuning fork — an acoustic resonator that vibrates/oscillates at a preferred frequency. This fundamental physical property of resonance selects inputs of preferred frequencies. This is a powerful signal detection technique employed by neurons that also serves as a bandpass filter, which blocks out non-preferred frequencies. Particularly for sensory coding, resonance can tune in preferred inputs.
Now we move on to biology. The physiology of a burst-encoding sensory neuron—which can behave like a resonator—consists of active and passive electrical properties, as well as the transmembrane gradients of ions like sodium (Na+) and potassium (K+). Passive electrical properties include the cell membrane’s capacitance and resistance. Active properties include the voltage-sensitive proteins (called voltage-gated ion channels), which are decked on the cell membrane. How do they work together? When a neuron receives external inputs from other neurons or peripheral receptors, the cell membrane’s voltage changes. The voltage-sensitive ion channels consequently undergo a conformational change (twist and bend) and behave like open gates, creating paths for ion flow in and out of the cell and in turn charging and discharging the capacitive membrane. Many types of ion channels have diverse properties, but we focus on the widely-present voltage-gated sodium channels (VGSCs).
Physiologists have long established that VGSCs are crucial for spike generation and information coding in neurons. These ion channels respond to a positive shift in membrane voltage called “depolarization” from a normal resting value of \(\sim -60\) mV. If the depolarization crosses a threshold value (\(\sim -40\) mV), VGSCs open and Na+ ions rush into the neuron that is driven by their concentration gradient. This causes further depolarization and opens more VGSCs, triggering more Na+ inrush (a positive feedback loop). Increased inrush of Na+ ions in turn prompts a steep upsurge in a neuron’s membrane voltage that results in an electrical impulse/spike. But this change is short-lived—lasting less than a millisecond—and the VGSCs “inactivate” quickly. Additionally, other voltage-gated ion channels (such as the K+ channels) open, letting K+ out of the cell along its own concentration gradient. This action forms a negative feedback loop, which turns off the electrical impulse, thus allowing the membrane voltage to return to the normal resting state within a few milliseconds. This is the basic phenomenology of spike generation. Next, I’ll briefly describe how VGSCs can also participate in the more exotic dynamics of burst generation.
VGSCs and Burst Generation
Once activated during a spike, the cumulative Na+ current via active VGSCs can exhibit multiple timescales of inactivation. A fast inactivation occurs within a single millisecond, while a slow component inactivates on a hundred-millisecond timescale. The exact biophysical mechanism of the slow component remains unclear. However, physiologists believe that it likely reflects an accumulation of activated VGSCs in inactivated states following each spike. This slow inactivation of VGSCs is crucial for burst generation. Following a burst, VGSCs are thought to recover back to a closed state and prepare for reactivation.
In a mathematical model for bursting, the Hopf bifurcation point of the fast spike generating subsystem by the slow Na+ inactivation/recovery variable—as the parameter in the singular limit—defines a threshold for burst initiation. The dynamics of a slow Na+ inactivation/recovery models the time course of slow inactivation and recovery of VGSCs. As Na+ channels slowly recover from inactivation during an IBI, the trajectory moves right (see Figure 1). Once the trajectory moves past the Hopf point, the membrane voltage undergoes a qualitative change from quiescence to the spiking phase. During a burst, the model Na+ current slowly inactivates and the trajectory moves left toward a burst termination threshold; this is due to a saddle-node of periodics (SNP) bifurcation.
VGSCs and Membrane Resonance
We now know that VGSCs can be more versatile. For instance, in combination with voltage-gated K+ channels and the plasma membrane’s passive capacitive property, a small percentage of VGSCs idiosyncratically remain open near the resting voltage of a neuron’s membrane. These properties result in small-amplitude subthreshold oscillations (STOs) of the neuronal membrane voltage. The STOs confer a natural frequency for a sensory neuron. When an input of preferred frequency comes along, the sensory neuron tunes in as a resonator and amplifies the membrane voltage response to such inputs (see Figure 2). Such input selectivity filter aids in signal detection and can encode the preferred input with features such as the ISIs and IBIs of a burst code.
VGSCs, Burst Stability, and Noise Modulation
During the period between bursts, spurious inputs can force abrupt transitions into the spiking phase and compromise a burst code’s information content. We tested whether Na+ currents can make a neuron tolerant of noise and showed that an additional Na+ component—mediated by the VGSCs—may play such a role. This unusual component—known as the resurgent Na+ current—was experimentally discovered in the sensory neurons under study, and shows unique properties in contrast to the fast and slow components described above. However, its role in burst coding is indeterminant due to a lack of suitable experimental approaches to study its individual contribution. We developed a novel mathematical model for the nonlinear Na+ current with resurgent component, which allowed us to study its exclusive role in burst encoding. Furthermore, we used a state-of-the-art real-time closed-loop experimental approach to interface the model with a live sensory neuron in a mouse brain slice, which validated the predictions.
The results were very intriguing. The resurgent Na+ current indeed peaks at the end of each spike; the resulting depolarization can regenerate spikes. This increases burst duration for a bursting neuron. Unexpectedly, we also found that this current, which is quite short lived (order of \(10\) ms), indeed affects the long-duration IBIs (order of \(100\) ms). Such IBI modulation filtered out noise-induced abrupt spikes and improved the regularity and entropy of bursting. Using bifurcation analysis, we showed that the resurgent Na+ can augment the slow inactivation of VGSCs, which in turn terminates a burst. This behaves like a negative feedback loop between the VGSC processes responsible for resurgent Na+ current and slow channel inactivation (see Figure 3). The phase plane in the figure demonstrates the region of attraction around the stable equilibria with and without the resurgent Na+ current, which is noise tolerant. As the trajectory courses along this region during an IBI (recovery of VGSCs from inactivation), the presence of resurgent Na+ and the enlarged separatrix makes the neuron less susceptible to random perturbations. Together with the burst-encoding tune-in filter, the resurgent mechanism helps tune out random noise following a burst.
Overall, our work provides insights into the neurocognitive mechanisms that make neurons good listeners (see Figure 4). Our mathematical model further motivates novel bioinspired filter design for signal detection and noise modulation in information processing systems.
Additionally, we utilized these models and bio-hybrid systems to study early disease dynamics in a neurodegenerative mouse model. We found that, these sensory neurons, which are muscle proprioceptors, show burst irregularities and reduction in Na+ currents during disease development. Real-time knock-in of the in-silico model Na+ currents into live sensory neurons restored burst regularity to control levels. Tracking such dynamic signatures of diseases using multiscale modeling and closed-loop experiments can guide the development of early biomarkers and neuro-rehabilitative technologies for devastating disabilities that result from neurodegeneration and neurotrauma.
This work was recently published in PLOS Computational Biology and the Journal of Neuroscience by the interdisciplinary team at University of California Los Angeles (UCLA), supported by the National Institute of Neurological Disorders and Stroke. Some of this work was also featured on ScienceTrends. The study was conducted by members of the Neural Dynamics Group in Scott Chandler’s lab, in collaboration with Riccardo Olcese’s lab at UCLA and David Terman at The Ohio State University.
||Sharmila Venugopal is an assistant adjunct professor in the Department of Integrative Biology and Physiology, Division of Life Sciences at the University of California, Los Angeles (UCLA). She is an electrical engineer and neuroscientist, and leads the Neural Dynamics Group at UCLA. Her research interests include computational neuroscience, neurophysiology and neural engineering with applications to health and disease. Venugopal also develops and teaches interdisciplinary math-biology courses for undergraduate life sciences majors at UCLA. She currently serves as a member of the Board of Directors and is Education and Training Chair for the Organization for Computational Neuroscience.