I completed my thesis in the Brain and Behaviour lab with Dr. Aldo Faisal.
Our nervous system, like most information processing systems, faces 4 fundamental physical constraints. It has to transmit information quickly (1. time) and reliably (2. noise) while keeping its energy consumption (3. energy) and size/weight (4. spatial scale) at a minimum, to support the behaviourally determined fitness of the organism. These constraints are likely to enforce trade-offs to be made in the evolution of the nervous system. Taking this view, I investigated myelinated and unmyelinated axons across systems and species. I developed user interfaces and simulation methods for the Modigliani stochastic simulation software, and made a number of findings highlighted in the following.
Myelinated axons make up the majority of long-range connections in CNS and PNS. I derived size-dependent relationships for metabolic costs of action potentials in myelinated axons. The high density of sodium channels at Nodes of Ranvier set lower limits on myelinated axons’ outer diameter (0.3 μm), which is 3-fold that for unmyelinated axons.
In contrast, thin, unmyelinated axons make up most of the local (cortical) connectivity. Using a variety of axon models and detailed models of synaptic calcium dynamics and vesicle release, I showed that the waveform of action potentials undergoes random changes whilst traveling along thin unmyelinated axons. These fluctuations translate into synaptic response variability. The diameter of unmyelinated axons sets energetic limits to signalling, and I derived diameter-dependent relationships for the maximum rate of burst and sustained firing. The latter depends on the density of pumps and metabolic cost of action potentials, but is counter intuitively independent of axon diameter.
My findings provide insights and scaling-relationships that relate the 4 fundamental design constraints for wiring brains. They allow us to quantitatively predict structure-function relationships, and form a basis for principled treatments of nerve disorders.
The action potential (AP), the fundamental signal of the nervous system, is carried by two types of axons: unmyelinated and myelinated fibers. In the former the action potential propagates continuously along the axon as established in large-diameter fibers. In the latter axons the AP jumps along the nodes of Ranvier—discrete, anatomically specialized regions which contain very high densities of sodium ion (Na+) channels. Therefore, saltatory conduction is thought as the hallmark of myelinated axons, which enables faster and more reliable propagation of signals than in unmyelinated axons of same outer diameter. Recent molecular anatomy showed that in C-fibers, the very thin (0.1 µm diameter) axons of the peripheral nervous system, Nav1.8 channels are clustered together on lipid rafts that float in the cell membrane. This localized concentration of Na+ channels resembles in structure the ion channel organization at the nodes of Ranvier, yet it is currently unknown whether this translates into an equivalent phenomenon of saltatory conduction or related-functional benefits and efficiencies. Therefore, we modeled biophysically realistic unmyelinated axons with both conventional and lipid-raft based organization of Na+ channels. We find that APs are reliably conducted in a micro-saltatory fashion along lipid rafts. Comparing APs in unmyelinated fibers with and without lipid rafts did not reveal any significant difference in either the metabolic cost or AP propagation velocity. By investigating the efficiency of AP propagation over Nav1.8 channels, we find however that the specific inactivation properties of these channels significantly increase the metabolic cost of signaling in C-fibers.
Post-synaptic potential (PSP) variability is typically attributed to mechanisms inside synapses, yet recent advances in experimental methods and biophysical understanding have led us to reconsider the role of axons as highly reliable transmission channels. We show that in many thin axons of our brain, the action potential (AP) waveform and thus the Ca++ signal controlling vesicle release at synapses will be significantly affected by the inherent variability of ion channel gating. We investigate how and to what extent fluctuations in the AP waveform explain observed PSP variability. Using both biophysical theory and stochastic simulations of central and peripheral nervous system axons from vertebrates and invertebrates, we show that channel noise in thin axons (<1 µm diameter) causes random fluctuations in AP waveforms. AP height and width, both experimentally characterised parameters of post-synaptic response amplitude, vary e.g. by up to 20 mV and 0.5 ms while a single AP propagates in C-fibre axons. We show how AP height and width variabilities increase with a ¾ power-law as diameter decreases and translate these fluctuations into post-synaptic response variability using biophysical data and models of synaptic transmission. We find for example that for mammalian unmyelinated axons with 0.2 µm diameter (matching cerebellar parallel fibres) axonal noise alone can explain half of the PSP variability in cerebellar synapses. We conclude that axonal variability may have considerable impact on synaptic response variability. Thus, in many experimental frameworks investigating synaptic transmission through paired-cell recordings or extracellular stimulation of presynaptic neurons, causes of variability may have been confounded. We thereby show how bottom-up aggregation of molecular noise sources contributes to our understanding of variability observed at higher levels of biological organisation.
Recent advances in experimental methods have allowed to reconsider the role of axons as faithful transmission channels. Post-synaptic potential (PSP) variability is typically attributed to mechanisms inside the synapse, yet in the many thin axons of our brain, the action potential (AP) waveform and thus the Ca++ signal controlling vesicle release at the synapse will be significantly affected by the inherent stochasticity of ion channels. We investigate to what extent fluctuations in the AP waveform cause the observed PSP variability. We show, using both biophysical theory and stochastic simulations of invertebrate and vertebrate, CNS and PNS axons, that channel noise in axons below 1 µm causes random changes in the AP waveform. AP height and width, both experimentally characterised parameters of post-synaptic response amplitude, vary e.g. by 6 mV and 1.5 ms as a single AP propagates in 0.3 µm diameter axons (cortical axon collaterals). We show how AP height and width variabilities increase with a 3⁄4 power-law as diameter decreases and translate these fluctuations into post-synaptic response variability using biophysical data and models of synaptic transmission. We find that e.g. in synapses innervated by 0.2 µm diameter unmyelinated axons (cerebellar parallel fibres) half of PSP variability can be accounted for by axonal noise alone. Axonal variability may have considerable impact on synaptic response variability. However, it can be easily confounded with synaptic variability in many experimental frameworks investigating synaptic transmission through paired-cell recordings or extracellular stimulation of the presynaptic neuron and intracellular post-synaptic recordings.
The energy necessary for propagating action potentials (APs) in axons is stored in the form of ionic concentration gradients across the membrane. It is commonly assumed that the number of ions crossing the membrane for each AP is very small compared to the total number of ions involved, as this is the case in classically studied axons e.g. the squid giant axon (SGA) with diameters of hundreds of microns. However, the mammalian nervous system contains much thinner axons e.g. C fibres or cortical axon collaterals (d=0.1-0.3 µm). The current due to ionic pumps is much smaller than that of ion channels in an action potential, which means that the potential energy pool may be depleted. We investigate how homeostatic and metabolic constraints limit neuronal activity using a Hodgkin-Huxley type model which tracks changes of ionic concentrations. A rough estimation yields a theoretical minimum of at least 0.3 mV of the potassium reversal potential (EK) in a C-fibre axon of diameter 0.3µm. Our simulations, which take into account the inefficiencies in channel kinetics, yield a higher value (1.2 mV) and show an inverse relationship between the change in reversal potentials and diameter. Based on basic physical considerations, we establish an equation describing the evolution of ionic concentrations and linking the maximum sustainable firing rate to the diameter. Using our model, we can predict the maximum firing rate that our model of the SGA can sustain for a given amount of time e.g. 10 seconds as a function of the axonal diameter. We conclude that in a similar way to noise, energetic considerations constrain the miniaturization of axons and limit the wiring density of neural circuits to energetically sustainable neural codes.
The fundamental signal of the nervous system, the action potential, is carried by two types of nerve fibres, myelinated and unmyelinated axons. Myelinated axons feature a highly structured distribution of voltage-gated ion channels, with a characteristic clustering of Na channels at the Nodes of Ranvier. In contrast, unmyelinated axons are generally thought to feature uniformly distributed ion channels. The former type of axon is generally known to allow faster, more reliable and, as we recently showed (Neishabouri & Faisal, 2011, BMC Neurosci), considerably more energy efficient propagation of signals than unmyelinated axons. In contrast, only the latter can reach the physical limits to axon diameter at 0.1 µm, thus making the high connection densities of mammalian cortex possible (Faisal et al., 2005, Curr Biol). In C-fibres, we have recently discovered that NaV1.8, the voltage-gated Na channels of these 0.1 µm diameter unmyelinated axons (Baker, 2005, J.Physiol), are packed tightly together on lipid rafts instead of being uniformly distributed over the membrane, resembling the structure of Nodes of Ranvier in myelinated fibres. Dissolving the lipid rafts in these axons can result in the loss of action potential transmission. We investigated the effects of the lipid-raft clustering of Na channels in terms of function, by looking at the propagation velocity, reliability and metabolic cost of action potentials. Because ion channel noise is significant in these axons we used the Modigliani stochastic simulation framework (Faisal & Laughlin, 2007, PLoS CB). We simulated both uniformly distributed and clustered channels (0.2µm diameter clusters of channels every 3µm of axon) along the unmyelinated fibre (0.1 µm diameter), so average channel density was in both cases 125 per µm\^2. Metabolic cost was defined by the amount of ATP molecules necessary to reverse the Na+ current by Na-K-ATPase (Alle et al., 2009, Science). We found that clustering Na channels on lipid raft reduced metabolic cost by over 260% (2.86×10\^-6 pmol ATP/mm vs. 1.03×10\^-5 pmol ATP/mm), while affecting propagation velocity and reliability far less (~20%). Thus, it appears that lipid rafts act like molecular Nodes of Ranvier in C-fibers, which as we show here is an evolutionary advantageous design feature that may be general to the many thin fibers of the CNS and PNS.