Homeostatic regulation of neuronal excitability



Homeostatic regulation of neuronal excitability refers to the collective phenomena by which neurons alter their intrinsic or synaptic properties to maintain a target level of electrical activity (see Fig. 1 & 2). Many types of homeostatic processes have been observed in neurons in different experimental preparations and contexts. Computationally, homeostatic plasticity is an important means of ensuring stability in network activity and electrical properties of individual units. This article focuses on homeostatic plasticity of single neurons, in particular, the activity-dependent regulation of ion channel densities that determine the intrinsic properties of neurons. The wider field of neuronal homeostasis covers many areas of neurophysiology, including structural plasticity, neurogenesis and cellular homeostasis.

History
The concept of homeostasis derives from the observation, due to Claude_Bernard Claude Bernard (1813-78) that organisms maintain some kind of constancy in their internal physiological environment (‘milieu interior’) in spite of external perturbations. This idea was later developed by Walter Cannon (1871-1945), who coined the term ‘homeostasis’ in his book The Wisdom of the Body (1932). Homeostasis was given a rigorous treatment and definition in the works of William_Ross_Ashby, Norbert Wiener and others in the field of Cybernetics which emerged in the 1940s-50s.

Until the 1990s the concept of homeostasis as it applies to neuronal synapses and intrinsic properties was absent from neurophysiology. At this time, evidence had accumulated that many neuronal properties such as synaptic strength and intrinsic excitability were not fixed, but exhibited activity-dependent plasticity. This plasticity was often characterized in terms of long-term potentiation or depression and interpreted as a Hebbian mechanism for storing memories. Several theoretical findings in the early 1990s pointed to an additional requirement for activity-dependent modification of intrinsic neuronal properties so that neurons can develop and maintain appropriate electrophysiological function. This non-Hebbian ‘homeostatic’ plasticity was also proposed as a means of regulating average activity in neuronal networks, thus preventing run-away excitation or quiescence that might occur if Hebbian mechanisms exist in isolation. These theoretical ideas were soon corroborated by experiments in vertebrate and invertebrate preparations that described activity-dependent changes in membrane conductances occurring over a slow timescale (hours to days) that tended to stabilize a particular activity regime such as rhythmic bursting.

Several years later, similar homeostatic processes were observed at chemical synapses, both in vitro and in vivo and at excitatory as well as inhibitory synapses. One important feature of synaptic homeostasis is scaling [see below] which ensures that relative synaptic strengths are preserved. As with intrinsic homeostasis, theoretical models of synaptic plasticity such as Oja’s rule (a mathematical formalization of Hebb's rule) anticipated synaptic homeostasis and its scaling property as a means of controlling Hebbian plasticity while preserving information stored in synaptic weight distributions.

Experimental findings
Neurons are long-lived, terminally differentiated cells (lasting many decades in humans) and have highly specialized electrical signaling properties that enable them to perform computations. Maintaining a stable electrical activity level over long time scales is a non-trivial problem because neuronal function is dependent on the expression of ion channels, which are short-lived and turn over in a stochastic manner. In addition, activity levels in an intact nervous system are subject to continual perturbations due to other processes including other forms of plasticity, environmental fluctuations and developmental changes. Homeostatic plasticity refers to cellular processes in neurons that serve to counter or compensate for such perturbations. Homeostatic effects have been characterized in many species and preparations including dissociated neuronal cultures, organotypic cultures and more recently in vivo. A summary of some major experimental findings over the past two decades is provided in Figure 3 (click to enlarge).

An early experiment showing intrinsic homeostatic plasticity was done by Franklin et al. (1990) in cultured rodent neurons. In this preparation, calcium currents were downregulated after neurons were depolarized for extended periods using high levels of extracellular potassium. This provided evidence that intrinsic conductances are subject to activity dependent regulation in a negative feedback loop that maintains a basal level of electrical activity and prevents cell death due to excitotoxicity. Later work in crustacean central pattern-generating neurons established that characteristic firing properties of neurons could be maintained in neurons by a feedback mechanism that likely uses intracellular calcium as a measure of ongoing activity. Many intrinsic currents appear to be subject to homeostatic plasticity, including \(Na^+\) currents, \(Ca^{2+}\) currents, \(K^+\) currents, and \(I_h\) - the hyperpolarization-activated cation current (see experiments in Figure 3).

Evidence for synaptic homeostatic plasticity has been gathered in systems including mammalian cortical neurons and the neuromuscular junction of Drosophila melanogaster. Manipulations that either increase or decrease synaptic activity are accompanied by alterations in synaptic strength over the course of several hours that counteract the changes in activity. Manipulations in vitro include the use of antagonists of excitatory and inhibitory synaptic transmission or pharmacological agents that increase or decrease intrinsic excitability of presynaptic neurons. Synaptic homeostasis has also been observed in vivo in response to changes in network activity due to sensory experience, pharmacological agents or genetic manipulations. Synaptic strength is correlated with the shape and size of pre-synaptic structures (such as the neuromuscular bouton) and post-synaptic structures (such as dendritic spines). This has enabled the characterization of homeostatic synaptic plasticity in terms of the quantity and size of these structures, which is an example of a more general phenomenon known as structural plasticity (see Kirov & Harris, 1999).

General theory
Abstractly, homeostatic plasticity is an example of feedback control. A canonical feedback control model (Figure 4) consists of a process (sometimes called the ‘system’ or ‘plant’) that transforms inputs into outputs. For example, a neuron transforms synaptic currents into action potentials. The process is driven toward some target output by a controller that takes output (or some function of output) and feeds a transformed version of this output signal back to the process. In order to implement this in a biological context, the control mechanism requires some kind of sensor of system output. Two important control-theoretic properties of a system are its observability and its controllability:


 * Observability measures how well the state of the system can be captured by sensor variables. An observable system is one in which the state can be completely and unambiguously determined by a set of output variables. In terms of neuronal homeostasis, observability could refer to the degree to which the voltage activity pattern of a neuron can be inferred from \(Ca^{2+}\) sensors or other putative biological sensors.


 * Controllability is a measure of the extent to which a system can be driven to a specified state by manipulating its input, for example, by tuning the parameters of the system. In neurons, this could refer to how many activity patterns can be produced by a controller that tunes the maximal conductances of ionic currents in a neuron. The set of outputs that can be achieved with a given controller is termed the reachable set, which is usually a subset of the system's entire state space.

Figure 4 shows the generic model of feedback control. This model captures specific theories of homeostatic plasticity as will be discussed in the next section. In the more general context of control theory, a number of standard controllers have been developed and applied to a wide variety of engineering applications. Several common examples are on-off controllers, proportional controllers, integral controllers, and PID controllers. As an example, we consider the operation of an on-off controller (or bang-bang controller). This is a simple feedback controller that switches its signal between two states around the set point. For example, an electric oven achieves a stable internal temperature by powering a heating element at full capacity until the oven thermometer senses that the set point is reached. At this point, the heating element turns off. If implemented in this simple form, a bang-bang controller would cause the heating coil to be repeatedly turned on and off as the oven temperature oscillates around the set point. In reality, this is avoided by introducing fixed delays in the controller.

Mathematically, we represent the influence of the controller on the system as a function \(u(t)\) and the error signal as \(s(t)-s_0\), where \(s(t)\) represents the sensor reading over time and \(s_0\) represents the target sensor value or set point (which we take to be constant). In terms of neuronal systems, \(u(t)\) can be viewed as the influence of the controller on the neural properties. For example, one could suppose that the maximal conductances (\(\bar{g}_i\)) all change over time according to \(d\bar{g}_i/dt = u(t) - \bar{g}_i\). Using this notation, a simple bang-bang controller can be represented as: \[u(t) = K_g H[s(t)-s_0]\text{,}\] where \(H[\cdot]\) represents the Heaviside step function, and the constant \(K_g\) represents the controller gain.

This simple example illustrates some of the key features of feedback control. Computation models of homeostatic plasticity are more specific to the problem of controlling neuronal activity by modulating ionic conductances and synaptic connectivity. Examples of such modeling work are discussed below.

Modeling work
Homeostatic plasticity is a specific form of feedback control that has been implemented in models of neurons and networks of neurons. The models described here are examples that were used to understand how neurons and networks with dynamic properties can maintain stable function. In keeping with the experimental distinction, these models can be broadly separated into those that regulate synaptic properties or intrinsic neuronal properties.


 * Models of synaptic homeostatic plasticity

Early computational models of neural networks achieved learning by updating synaptic weights according to a “learning rule” (see Hebbian Theory, and Perceptron). In isolation, Hebbian mechanisms can be unstable because they tend to generate positive feedback. Therefore many learning rules implemented versions of Hebbian learning with some kind of synaptic weight normalization or other scheme to control weight increase. Oja’s rule, which re-normalizes the synaptic weights as they are updated, and BCM theory, in which synaptic potentiation saturates as post-synaptic activity increases, were developed (among others) to provide stability to synaptic learning and to explain experimentally-observed competitive processes that appeared to counter unstable growth of synaptic strength. This work was important for experimental studies because theoretical models of this kind predicted that classical Hebbian learning could not be implemented in a biological system without some kind of modification or additional homeostatic process.


 * Models of intrinsic homeostatic plasticity

The first model of intrinsic homeostatic intrinsic plasticity was developed by LeMasson et al. (1993) and Abbott and LeMasson (1993). This model was inspired by the observation that in crustacean rhythm-generating circuits, neurons appear to maintain characteristic electrical activity. It was proposed that neurons have a built-in sensor mechanism that monitors electrical activity and adjusts conductance densities to maintain a 'target' activity level. A biologically plausible sensor mechanism that subsequently found experimental support (see section on experimental findings) was postulated to be some kind of calcium-sensitive signalling process that is sensitive to fluctuations in calcium concentration due to activity over extended periods. In the LeMasson-Abbott model, maximal conductance are varied as a function of intracellular $$Ca^{2+}$$ according to the following equations:

\[ \tau \frac{d \bar{g}_i}{dt} = f_i([Ca]) - \bar{g}_i   \hspace10ex    \text{where,}\hspace1ex f_i([Ca]) = \frac{G_i}{1+\exp\{\pm ([Ca]-C_T)/\Delta\}}\]

Where \(\tau\) is the time constant for maximal conductance regulation (picked to be slower than membrane potential dynamics), \(\bar{g}_i\) is the maximal conductance for the \(i^{th}\) ionic current (for \(n\) currents, \(i=\{1,2,...,n\}\)), \(f_i\) is a sigmoidal activation function, \([Ca]\) is the intracellular calcium concentration, \(G_i\) is the maximum value of the maximal conductance for the \(i^{th}\) current,\(C_T\) is the target intracellular calcium level, and \(\Delta\) controls the slope of the sigmoidal curve. The parameters in this model were chosen such that intracellular $$Ca^{2+}$$ approached a specified target concentration. The end result was a model neuron that produced periodic bursts of action potentials, and was able to compensate for perturbations such as changes to the $$K^+$$ reversal potential.

An important feature of the LeMasson model is that it converges to the same final set of maximal conductances, regardless of the initial set of conductances at the beginning of the simulation. A subsequent model by Liu et al. (1998) produced variable sets of maximal conductances, in line with subsequent experimental findings. The Liu model added further flexibility by introducing the idea that a neuron may have multiple homeostatic mechanisms operating with different types of activity sensor. Each conductance in the Liu model is regulated by three $$Ca^{2+}$$-dependent “sensors” which activate and inactivate on different time scales. The activity level of each sensor is represented by a dynamic variable (\(F\),\(S\), and \(D\); for "fast","slow", and "DC"). The differential equations governing these variables are very similar to channel kinetics in Hodgkin-Huxley type models. Each maximal conductance is regulated according to:

\[ \tau \frac{d \bar{g}_i}{dt} = [ A_i(\bar{F}-F) + B_i(\bar{S}-S) + C_i(\bar{D}-D)] \bar{g}_i \]

Where \(\tau\) is the time constant for maximal conductance regulation, \(\bar{g}_i\) is the maximal conductance for the \(i^{th}\) ionic current, and \(\bar{F}, \bar{S}, \bar{D}\) represent the target activity levels for each sensor. \(A_i, B_i, \text{and}\hspace1ex C_i\) each represent the direction of conductance regulation for the \(i^{th}\) current - they each can take on one of three values \(\{-1,0,+1\}\), which respectively represent that the conductance is up-regulated, not regulated, or down-regulated when sensor activity is below target activity. Figure 5 shows the behavior of the Liu model in response to a perturbation of hyperpolarizing current.

Later studies investigated the role of intrinsic homeostatic plasticity in shaping the computational properties of neurons. Stemmler and Koch (1999) developed a model in which a neuron is homeostatically tuned to maximize information transfer. Thus, rather than having a pre-defined ‘target’ activity pattern, the model changes its activity pattern based on the synaptic inputs that it receives, to maximize the mutual information between its synaptic inputs and its firing rate. Triesch (2005, 2007) derived an intrinsic plasticity rule based on information-maximization principles and uncovered an interaction between Hebbian learning and intrinsic homeostasis that permitted learning with stimulus statistics for which Hebbian mechanisms alone fail.

Experimental
At the time of writing, relatively little is known about the molecular signaling mechanisms that underlie homeostatic plasticity. The identity of the activity sensors and their downstream targets is unknown and the means by which trafficking or modification of ion channel and synaptic receptors occurs has not been fully characterized. One approach to this problem has been to use a genetic screen to identify mutants that are unable to homeostatically compensate to known perturbations (e.g. Frank et al., 2006). Additionally, little is known about how homeostatic compensation varies across cell types, or across different developmental stages of the same cell type. A large number of experiments have been done in cultured cortical neurons from prenatal or very young mice. This raises the question as to whether some of the compensatory responses observed in these preparations indicate ‘developmental robustness’ rather than ongoing homeostasis in adult nervous systems.

Theoretical
Previous theoretical work has developed a number of models of neuronal homeostasis. Because there are many different models that can achieve stable activity, a central problem is to characterize these different models in terms of their testable predictions. For example, the abstract on-off, proportional, and integral controllers all produce different dynamical properties. Likewise, the controller mechanisms that underlie homeostasis in biological neurons will have characteristic dynamics that lead to experimentally testable predictions. The form of these biological controllers has not yet been uncovered and the dynamics of homeostasis been yet to be described in full detail. Additionally, while neuronal homeostasis has been incorporated into some network models (for example, Soto-Trevino et al., 2001; Meltzer et al., 2005; Remme & Wadman, 2012), an understanding of network homeostasis is far from complete. Specifically, it is not known how homeostatic processes in individual neurons interact on the network level. Similarly, it is not fully known how homeostatic regulation at multiple sites (synaptic vs intrinsic properties) and multiple timescales (minutes, hours and days) interact within neurons and with other plasticity mechanisms.

Selected Reviews

 * Queenan BN, Lee KJ, Pak DTS (2012). "Wherefore art thou, Homeo(stasis)? Functional diversity in homeostatic synaptic plasticity." Neural Plasticity. vol. 2012, Article ID: 825364, 12 pages. doi:10.1155/2012/718203
 * Wang G, Gilbert J, Man H (2012). "AMPA Receptor Trafficking in Homeostatic Synaptic Plasticity: Functional Molecules and Signaling Cascades." Neural Plasticity. vol. 2012, Article ID: 825364, 12 pages. doi:10.1155/2012/825364
 * Turrigiano G (2011). "Too many cooks? Intrinsic and synaptic homeostatic mechanisms in cortical circuit refinement." Annu Rev Neurosci. 34:89-103.
 * O'Leary T & Wyllie, DJA (2011). Neuronal homeostasis: time for a change? Journal of Physiology 589, 4811-4826.
 * Pozo K, Goda Y (2010). "Unraveling mechanisms of homeostatic synaptic plasticity." Neuron. 66(3):337-51.
 * Turrigiano GG (2008). "The self-tuning neuron: synaptic scaling of excitatory synapses." Cell. 135(3):411-35.
 * Nelson SB, Turrigiano GG (2008). "Strength through diversity." Neuron. 60(3):477-82.
 * Marder E, Goaillard JM (2006). "Variability, compensation and homeostasis in neuron and network function." Nature Rev Neurosci. 7:563-74.
 * Davis GW (2006). "Homeostatic control of neural activity: from phenomenology to molecular design." Annu Rev Neurosci. 29:207-23.
 * Desai, N. S. (2003). Homeostatic plasticity in the CNS: synaptic and intrinsic forms. Journal of Physiology, Paris. 97(4-6), 391.
 * Burrone J, Murthy VN (2003). "Synaptic gain control and homeostasis." Curr Opin Neurobiol. 13(5):560-67.
 * Turrigiano, GG, & Nelson, SB (2000). Hebb and homeostasis in neuronal plasticity. Current Opinion in Neurobiology. 10(3), 358-364.
 * Abbott LF, Nelson SB (2000). "Synaptic Plasticity: Taming the Beast." Nature Neurosci. 3:1178-83.

Selected Primary Research Papers

 * Mitra A, Mitra SS, Tsien RW (2012) "Heterogeneous reallocation of presynaptic efficacy in recurrent excitatory circuits adapting to inactivity." Nat Neurosci. 15: 250–257.
 * Jakawich SK, Nasser HB, Strong MJ, McCartney AJ, Perez AS, Rakesh N, Carruthers CJ, Sutton MA (2010) "Local Presynaptic Activity Gates Homeostatic Changes in Presynaptic Function Driven by Dendritic BDNF Synthesis". Neuron. 68(6): 1143-1158.
 * Olypher AV, Prinz AA (2010). "Geometry and Dynamics of Activity-Dependent Homeostatic Regulation in Neurons." J Comp Neurosci. 28:361-374.
 * Gunay C, Prinz AA (2010). "Model calcium sensors for network homeostasis: sensor and readout parameter analysis from a database of model neuronal networks." J Neurosci. 30:1686-98.
 * Triesch J (2005). “A gradient rule for the plasticity of a neuron's intrinsic excitability.” Artificial Neural Networks: Biological Inspirations. Proceedings of the International Conference on Artificial Neural Networks. 3696:65-70.
 * Yeung LC, Shouval HZ, Blais BS, Cooper LN (2004). "Synaptic homeostasis and input selectivity follow from a calcium-dependent plasticity model." Proc Natl Acad Sci U S A. 101:14943-48.
 * Giugliano M, Bove M, Grattarola M (1999). "Activity-driven computational strategies of a dynamically regulated integrate-and-fire model neuron." J Comput Neurosci. 7:247-254
 * Golowasch J, Casey M, Abbott LF, Marder E (1999). "Network stability from activity-dependent regulation of neuronal conductances." Neural Comput. 11(5):1079-96
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