Current Projects
Dynamic Clamp Accuracy: Time Step, Jitter, and Latency
An experiment can only be as accurate as the tools it uses. A dynamic clamp experiment uses a computer to make rapid calculations in order to manipulate real neurons. The delicate electrical activity of neurons happens on a microsecond timescale, and thus a good experiment requires a computer with a high degree of temporal precision.
NDL has begun to characterize the amount of "wiggle room" a dynamic clamp computer has before it starts producing inaccurate neuron behavior that affects experimental results.
Background
Neurons communicate through electrochemical signals called action potentials. The inside of a neuron at rest has an electrical potential that is slightly negative relative to the outside, which creates a voltage drop across the cell membrane of about -65 mV. Stimulation by a small electrical current produces action potentials in the neuron, i.e. the membrane potential (VM) undulates. (See Fig. 1) It is important to note that the amplitude of the action potentials is constant, although their frequency is dependent on the magnitude of the stimulating current.
Figure 1: A neuron
firing action potentials
Action potentials are created primarily by channels that regulate the movement of electrically charged sodium and potassium ions (Na+ and K+) through the cell membrane. The movement of these ions creates an electrical current that changes the membrane potential. The membrane potential, in turn, regulates the activity of the ion channels.
In order to precisely control a neuron, one must be able to calculate the current flowing through the Na and K channels. The Hodgkin-Huxley model is one solution that represents the membrane as a circuit where the Na and K channels are voltage-dependant conductances. (See Fig. 2) The model we use for our error characterization experiments is slightly more complex, but it has the same basic structure.
Figure 2: The Hodgkin-Huxley circuit
model
In dynamic clamp experiments, a computer monitors the membrane voltage and injects a very specialized current into a neuron. Essentially, the computer "pretends" to be either the Na or the K channels and injects the current that the movement of ions through those channels would otherwise be providing.
For example, suppose we wish to virtualize the K channels. First, the real K channels in a neuron are knocked out with a neurotoxin. The handicapped neuron can no longer produce action potentials on its own because it lacks the current from the K channels. The dynamic clamp reads the membrane potential and uses it to calculate what electrical current the potassium channels would be supplying if they were active. It uses this information to guesstimate what that current will be one time step ahead in the future and then injects that current back into the cell. (See Fig. 3)
Figure 3:
Dynamic clamp schematic
It is important to stress that this current is dependent on the membrane voltage. When the voltage changes, it affects the current, which in turn changes the voltage, which changes the current, and so and so forth. Thus, the current provided by the K channels is not constant, but has a special "shape." If the computer is a good dynamic clamp, it will mimic this shape very accurately, and the neuron will be able to fire action potentials again with the help of the computer.
Error Characterization
Neurons are very sensitive
to the current shape from their ion channels. If the dynamic clamp
computer is slow and doesn't read the neuron quickly enough, it injects a
funny-shaped current which causes the neuron to react in an unnatural way, which
can ruin an experiment. For example, notice the blocky shape of the current in
Fig 3. The faster the computer is, the smoother this curve becomes, and the more
closely it resembles the natural curve from the neuron. The resolution of the
curve is dependent upon the time step of the dynamic clamp, which is the amount
of time it takes between calculations. If the time step is too large, the action
potentials loose accuracy. For example, in Fig. 4, a time step of 25
microseconds (40 kHz) yields action potentials that are nearly indistinguishable
from those of a healthy neuron. Here, an action potential is roughly 6
milliseconds long, so the dynamic clamp is reading the neuron 240 times during
each peak. However, action potentials created with a 200 microsecond time step
(30 times per peak, or 5 kHz) are incredibly inaccurate.
Figure 4: Action potentials with different dynamic clamp time
steps.
In order to qualify how inaccurate an action potential is, the squared error is calculated from the difference between the two curves after their front edges have been aligned. Fig 5 shows the error curves for virtualized Na and K simulations. Notice that virtualized K error is generally less than Na error for any give time step. Also, single virtualized Na action potentials begin showing double peaks (i.e. like the red plot in Fig 4) around 0.125 ms (8 kHz).
Figure 5
However, it is also very likely that a dynamic clamp will not be able to maintain a constant time step and will have some jitter. For example, a simulation with no jitter would consistently take a time step of 0.100 ms, but a simulation with jitter might take a time step at 0.075 ms, then 0.130 ms, 0.090 ms, etc. Another factor is that the clamp may not be aware of the actual time step it's using and instead uses a constant time step to make its calculations. Fig. 6 shows the effects of jitter, where "smart jitter" is when the clamp knows what the time step is and "blind jitter" is when it does does not. Notice that after about 0.125 ms (8 kHz), the no jitter, smart jitter, and blind jitter plots become increasingly indistinguishable from one another.
Another way the dynamic clamp can induce error is through latency, which is the time lag between when the computer calculates the current and when the current is applied. Intuitively, the more latency a system has, the greater the error. However, it also appears that latency tends to pull K error above Na error. (Remember that K error was below Na error in Fig. 5?)
Figure 7: Virtualized Na and K error with a
latency of 50% of the time step.
All of the above figures have only addressed the accuracy of the shape of individual action potentials, but have neglected the space between peaks or the frequency at which they occur. Preliminary experiments seem to indicate that timestep, jitter, and latency have relatively little effect on frequency.
© Laura Stupin 2005
Resources
Migliore, Hoffman, et. al (1999) Role of an A-Type K+ Conductance in the Back-Propagation of Action Potentials in the Dendrites of Hippocampal Pyramidal Neurons. Journal of Computational Neuroscience.
Prinz AA, Abbott LF, Marder E (2004) The dynamic clamp comes of age. Trends Neurosci. 27(4):218-24.
White JA (2001) Action Potential. In Encyclopedia of the Human Brain. V.S. Ramachandram (Ed.), Academic Press, San Diego, in press. [pdf]
Channel Noise and Neuronal Dynamics
Neuronal activity is characterized by short action potentials generated and triggered by various voltage gated ion channels. Nonlinear systems like this, whose components may act over multiple time scales, are extremely sensitive to noise. We are investigating how the interaction of ion channel noise and a neuron's internal dynamics can affect the neuron's expressible behavior in three different systems.
In response to constant current, stellate cells of the medial entorhinal cortex generate clusters of spikes that are loosely phase-locked to a subthreshold voltage oscillation. This unusual pattern of activity is associated with a high variability in spike intervals. We find that it develops in the juvenile between days 14 and 18 after birth and we are currently investigating its ionic basis. Among other possibilities, a likely candidate mechanism is the expression of a limited number of persistent sodium channels, known to be required for the oscillation in the adult. In computational and electrophysiological work in mature rats, we have examined how channel noise associated with these channels affects neuronal behavior. Our findings suggest that channel noise promotes spontaneous activity, softens action potential thresholds and is essential for clustered spiking.
The firing patterns of auditory nerve fibers are random in response to normal synaptic input, as well as in response to direct electrical input when delivered at sufficiently low levels. In another project, we apply mathematical models to elucidate the roles of physiological noise and neuronal dynamics in generating this random behavior.
Additional modeling studies are being conducted in collaboration with the laboratory of Simon Laughlin at the University of Cambridge. In this work, led by Cambridge graduate student A. Aldo Faisal, we argue that channel noise may be a crucial factor in limiting minimal axonal diameters, and thus the maximal degree of interconnectivity, in the nervous system.
Recent Conference Abstracts:
Burton BG and White JA (2004). Development of oscillatory behaviour in stellate cells of the rat entorhinal cortex. IVth Forum of the Federation of European Neuroscience Societies, Lisbon.
Related publications
Faisal AA, Laughlin SB, and White JA (2002) How reliable is the connectivity in cortical neural networks? Proceedings of the 2002 International Joint Conference on Neural Networks 2: 1661-1666.
White JA and Haas JS (2001) Noise from voltage-gated ion channels: effects on dynamics and reliability in intrinsically oscillatory neurons. In Handbook of Biological Physics, Vol. 4. F Moss and S Gielen (Eds.), Elsevier Press. [pdf]
White JA, Rubinstein JT, and Kay AR (2000) Channel noise in neurons. Trends in Neurosciences 23: 131-137. [abstract] [pdf]
Lowen, SB, Liebovitch, L, and White JA (1999) Fractal ion-channel behavior generates fractal firing patterns in neuronal models. Physical Review E 59: 5970-5980. [abstract] [pdf]
Klink R, Alonso A, and Kay AR (1998) Noise from voltage-gated ion channels may influence neuronal dynamics in the entorhinal cortex. Journal of Neurophysiology 80: 262-269. [abstract] [full text] [pdf]Chow CC and White JA (1996) Spontaneous action potentials due to channel fluctuations. Biophysical Journal 71: 3013-3021. [abstract]
Reliable Responses in the Entorhinal Cortex
We apply our dynamical-systems and engineering toolboxes to excitatory cells in the superficial medial entorhinal cortex (MEC). In particular, we examine stellate cells (SCs) from layer II of MEC. These cells comprise the majority of cells in layer II ( Klink & Alonso, 1997 ) and their axons contribute substantially to the perforant path, the major source of input to the hippocampus. In response to constant depolarizing input, stellate cells show prominent ~8-Hz subthreshold oscillations and/or phase-locked action potentials that are hypothesized to contribute to the hippocampal theta rhythm.
Currently we are investigating what aspects of their inputs cause reliable firing in SCs. For example, SCs respond more reliably to inputs with the majority of their energy in the 4-12 Hz theta band. Also the reliability of theta frequency spikes increases with overall input conductance. These results suggest that SCs are well suited to respond to inputs rich in the frequencies seen spontaneously in response to constant current and seen in population activity during exploratory behavior. Nevertheless, even in reponse to robust theta sinusoidal input, SCs exhibit variability in their spike output. The patterns of spikes we observe indicate that a short term memory for its own recent activity influences spike generation up to 1 second into the future. Using a statistical approach, we are currently investigating to what extent this memory can affect the output patterns of the cell in response to simple modulated input.
Recent Conference Abstracts:
Burton BG and White JA (2003) The role of cellular memory in the function of stellate cells in the rat entorhinal cortex. Society for Neuroscience, 171.12. [pdf]
Dorval AD and White JA (2003) Stellate cell responses to conductance input: reliability resonance in the theta frequency band. Society for Neuroscience, 258.5. [pdf]
Haas JS, Dorval AD, and White JA (2002) Resonance and feature selectivity in stellate cells of the entorhinal cortex. Society for Neuroscience, 753.9. [pdf]
Related Publications:
Dorval AD, Netoff TI, and White JA (2003) Real-time experimental control in cellular neurophysiology. Proceedings of the 1 st International IEEE EMBS Conference on Neural Engineering, 71-74. [pdf]
Haas JS and White JA (2002) Frequency selectivity of layer II stellate cells in the medial entorhinal cortex. Journal of Neurophysiology 88(5): 2422-2429. [abstract] [pdf]
White JA and Haas JS (2001) Noise from voltage-gated ion channels: effects on dynamics and reliability in intrinsically oscillatory neurons. In Handbook of Biological Physics, Vol. 4. F Moss and S Gielen (Eds.), Elsevier Press. [pdf]
White JA, Budde T, and Kay AR (1995) A bifurcation analysis of neural subthreshold oscillations. Biophysical Journal 69: 1203-1217. [abstract]
Synchrony in Neuronal Networks
Synchronous (i.e., simultaneous) activity among neurons arises rather commonly and is important for cognitive function. In general, the behavior of a network of neurons and their ability to synchronize depends on four factors: the pattern connectivity within the network, the details of chemical and electrical connections between neurons, the intrinsic properties of the individual neurons (e.g., ionic conductances, dendritic morphology), and external modulation or drive of the network. Synchronous activity has been a consistent theme in much of our work. Completed and ongoing projects include the following:
In related theoretical and computational studies (Chow et al. 1998, White et al. 1998), we investigated synchronization in heterogeneous networks of inhibitory interneurons. We found that such networks generate robust synchronous activity only when the frequency of the oscillation matches the decay time constant of the inhibitory conductances. We later extended this approach to show that networks containing multiple, known populations of interneurons, giving rise to rapidly- and slowly-decaying inhibition, can support simultaneous theta and gamma rhythms (White et al. 2000).
We have adapted spike time response techniques for use in electrophysiology. These techniques allow one to predict the synchronization properties of a pool of neurons based on rather simple measurements of how synaptic inputs change spike timing. In modeling work (Acker et al. 2003), we have found that STR relationships change nonlinearly for some cells with physiologically realistic synaptic inputs. This nonlinearity complicates matters, but we are still able to measure STR relationships in living neurons (Netoff et al. 2003).
We have tested predictions from STR theory by building "hybrid" networks, consisting of multiple biological and virtual neurons, coupled in real time by artificial synapses. We find that predictions from simple STR measurements hold up remarkably well in free-running hybrid networks. Surprisingly, given modeling results (Acker et al. 2003), synchronization properties for identified cellular populations are remarkably consistent across the population, and largely independent of synaptic amplitudes and kinetics (Netoff et al. 2003). We are currently extending this approach to account for larger, more realistic neuronal networks.
Epilepsy is characterized by two types of neuronal activity: (1) interictal bursts, short bursts of large electrical activity in the brain lasting 100 ms; and (2) seizures, bouts of large electrical activity lasting on the order of seconds to minutes. These electrical activities are thought to be generated by hyper-synchronous neuronal activity. We have hypothesized that bursts are in fact synchronous events of neuronal activity, but seizures, in order to sustain long durations of activity, are less synchronous. We have explored this hypothesis further in neural network models with varied connectivity. We have found that long-distance connections help synchronize the network. As we vary the number of long distance connections in the network, we can transition the behavior from "normal" to "seizing" to "bursting." These behaviors are well described by a simple "birth and death" process model (Clewley et al. 2003).
Recent Conference Abstracts:
Clewley R, Netoff T.I., Arno A., Keck T, White JA. The role of network connectivity in explaining transitions to and from bursting in epileptiform and developing networks: a mathematical model. Society for Neuroscience Abstracts 31: 411.14. [pdf]
Netoff TI, Banks MI, White JA. (2003) Bridging single cell and network dynamics. Society for Neuroscience Abstracts 31: 171.7. [pdf]
Related Publications:
Acker, C. D., N. Kopell , et al. (2003). Synchronization of strongly coupled excitatory neurons: relating network behavior to biophysics. J Comput Neurosci 15 : 71-90. [abstract] [pdf]
Chow CC, White JA, Ritt J, and Kopell N (1998) Frequency control in heterogeneous inhibitory networks. Journal of Computational Neuroscience 5 : 407-420. [abstract] [pdf]
White JA, Chow CC, Ritt J, Soto-Treviño C, and Kopell N (1998) Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons. Journal of Computational Neuroscience 5 : 5-16. [abstract] [pdf]
White JA, Banks MI, Pearce, RA, and Kopell N (2000) Networks of interneurons with fast and slow GABA A kinetics provide substrate for mixed gamma-theta rhythm. Proceedings of the National Academy of Sciences USA 97 : 8128-8133. [abstract] [pdf]
Active Dendrites
Recent advances in experimental techniques have allowed researchers to focus on the electrophysiological properties of dendrites. Much of this work focuses on active properties of dendrites, which allow the backpropagation of somatic action potentials. Two ongoing projects in the lab focus on active dendrites:
In CA1 pyramidal cells, we have extended work on synchrony in neuronal networks to account for dendritic inputs. In simplified models of pyramidal cells, timing responses only changed in amplitude as a function of distance from soma; the shapes of spike time response curves (STRCs), and thus the predicted synchronization properties of the cells, remained unchanged (Acker et al. 2003). We see contrasting results in experiments. STRCs generated using glutamate application were rather uniform along the dendrite (40-200 m m from soma) but differed significantly with those measured at the soma (Keck et al. 2003).
We have begun testing the self-consistency of simple models of spike-time-dependent plasticity (STDP). We find that straightforward applications of these models cannot account for important features of STDP. Specific nonlinear modifications of the models mitigate this problem (Acker et al. 2003).
Recent Conference AbstractsAcker CD, Sen K, White JA (2003) Backpropagating action potentials, role in neural synchronization and synaptic plasticity. Society for Neuroscience Abstracts 31 : 257.17. [pdf]
Keck T, Netoff TI, White JA, (2003) Comparing dynamic responses of neurons to dendritic and somatic inputs. Society for Neuroscience Abstracts 31 : 476.7.