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Stability of Neuronal Networks with Homeostatic Regulation.

Harnack D, Pelko M, Chaillet A, Chitour Y, van Rossum MC - PLoS Comput. Biol. (2015)

Bottom Line: Next, we consider how non-linearities in the neural activation function affect these constraints.Finally, we consider the case that homeostatic feedback is mediated via a cascade of multiple intermediate stages.Counter-intuitively, the addition of extra stages in the homeostatic control loop further destabilizes activity in single neurons and networks.

View Article: PubMed Central - PubMed

Affiliation: School of Informatics, University of Edinburgh, Edinburgh, United Kingdom.

ABSTRACT
Neurons are equipped with homeostatic mechanisms that counteract long-term perturbations of their average activity and thereby keep neurons in a healthy and information-rich operating regime. While homeostasis is believed to be crucial for neural function, a systematic analysis of homeostatic control has largely been lacking. The analysis presented here analyses the necessary conditions for stable homeostatic control. We consider networks of neurons with homeostasis and show that homeostatic control that is stable for single neurons, can destabilize activity in otherwise stable recurrent networks leading to strong non-abating oscillations in the activity. This instability can be prevented by slowing down the homeostatic control. The stronger the network recurrence, the slower the homeostasis has to be. Next, we consider how non-linearities in the neural activation function affect these constraints. Finally, we consider the case that homeostatic feedback is mediated via a cascade of multiple intermediate stages. Counter-intuitively, the addition of extra stages in the homeostatic control loop further destabilizes activity in single neurons and networks. Our theoretical framework for homeostasis thus reveals previously unconsidered constraints on homeostasis in biological networks, and identifies conditions that require the slow time-constants of homeostatic regulation observed experimentally.

No MeSH data available.


Related in: MedlinePlus

Single neuron homeostasis.A) Schematic illustration of the homeostatic model. The input current is transformed through an input-output relation and a filter. The input-output curve is shifted by a filtered and integrated copy of the output firing rate, so that the average activity matches a preset goal value. F1 (time-constant τ1) denotes a filter describing the filtering between input and output of the neuron; F2 (time-constant τ2) is a filter between the output and the homeostatic controller. B) The response of the model for various settings of the homeostatic time-constants. The value of τ1 was fixed to 10ms (thin lines), while τ2 and τ3 were varied. Center plot: the response of the neuron can either be stable (top left plot; white region), a damped oscillation (top right plot, gray region), or unstable (bottom right plot, striped region). The surrounding plots show the firing rate of the neuron and the threshold setting in response to a step stimulus.
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pcbi.1004357.g001: Single neuron homeostasis.A) Schematic illustration of the homeostatic model. The input current is transformed through an input-output relation and a filter. The input-output curve is shifted by a filtered and integrated copy of the output firing rate, so that the average activity matches a preset goal value. F1 (time-constant τ1) denotes a filter describing the filtering between input and output of the neuron; F2 (time-constant τ2) is a filter between the output and the homeostatic controller. B) The response of the model for various settings of the homeostatic time-constants. The value of τ1 was fixed to 10ms (thin lines), while τ2 and τ3 were varied. Center plot: the response of the neuron can either be stable (top left plot; white region), a damped oscillation (top right plot, gray region), or unstable (bottom right plot, striped region). The surrounding plots show the firing rate of the neuron and the threshold setting in response to a step stimulus.

Mentions: To examine the stability of homeostatic control we first analyze a single neuron with homeostasis, a schematic is shown in Fig 1A. We describe the activity of the neuron as a function of time with a firing rate r1(t). A common approximation for the firing rate dynamics isτ1dr1(t)dt=-r1(t)+g(u(t)-θ(t))(1)which can be understood as follows: The time-constant τ1 determines how rapidly the firing rate changes in response to changes in the input and how rapidly it decays in the absence of input. We use τ1 = 10 ms. The value of τ1 serves as the time-constant with respect to which all the other time-constants in the system will be defined. As only the ratios between time-constants will matter, the results are straightforwardly adapted to other values of τ1.


Stability of Neuronal Networks with Homeostatic Regulation.

Harnack D, Pelko M, Chaillet A, Chitour Y, van Rossum MC - PLoS Comput. Biol. (2015)

Single neuron homeostasis.A) Schematic illustration of the homeostatic model. The input current is transformed through an input-output relation and a filter. The input-output curve is shifted by a filtered and integrated copy of the output firing rate, so that the average activity matches a preset goal value. F1 (time-constant τ1) denotes a filter describing the filtering between input and output of the neuron; F2 (time-constant τ2) is a filter between the output and the homeostatic controller. B) The response of the model for various settings of the homeostatic time-constants. The value of τ1 was fixed to 10ms (thin lines), while τ2 and τ3 were varied. Center plot: the response of the neuron can either be stable (top left plot; white region), a damped oscillation (top right plot, gray region), or unstable (bottom right plot, striped region). The surrounding plots show the firing rate of the neuron and the threshold setting in response to a step stimulus.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4495932&req=5

pcbi.1004357.g001: Single neuron homeostasis.A) Schematic illustration of the homeostatic model. The input current is transformed through an input-output relation and a filter. The input-output curve is shifted by a filtered and integrated copy of the output firing rate, so that the average activity matches a preset goal value. F1 (time-constant τ1) denotes a filter describing the filtering between input and output of the neuron; F2 (time-constant τ2) is a filter between the output and the homeostatic controller. B) The response of the model for various settings of the homeostatic time-constants. The value of τ1 was fixed to 10ms (thin lines), while τ2 and τ3 were varied. Center plot: the response of the neuron can either be stable (top left plot; white region), a damped oscillation (top right plot, gray region), or unstable (bottom right plot, striped region). The surrounding plots show the firing rate of the neuron and the threshold setting in response to a step stimulus.
Mentions: To examine the stability of homeostatic control we first analyze a single neuron with homeostasis, a schematic is shown in Fig 1A. We describe the activity of the neuron as a function of time with a firing rate r1(t). A common approximation for the firing rate dynamics isτ1dr1(t)dt=-r1(t)+g(u(t)-θ(t))(1)which can be understood as follows: The time-constant τ1 determines how rapidly the firing rate changes in response to changes in the input and how rapidly it decays in the absence of input. We use τ1 = 10 ms. The value of τ1 serves as the time-constant with respect to which all the other time-constants in the system will be defined. As only the ratios between time-constants will matter, the results are straightforwardly adapted to other values of τ1.

Bottom Line: Next, we consider how non-linearities in the neural activation function affect these constraints.Finally, we consider the case that homeostatic feedback is mediated via a cascade of multiple intermediate stages.Counter-intuitively, the addition of extra stages in the homeostatic control loop further destabilizes activity in single neurons and networks.

View Article: PubMed Central - PubMed

Affiliation: School of Informatics, University of Edinburgh, Edinburgh, United Kingdom.

ABSTRACT
Neurons are equipped with homeostatic mechanisms that counteract long-term perturbations of their average activity and thereby keep neurons in a healthy and information-rich operating regime. While homeostasis is believed to be crucial for neural function, a systematic analysis of homeostatic control has largely been lacking. The analysis presented here analyses the necessary conditions for stable homeostatic control. We consider networks of neurons with homeostasis and show that homeostatic control that is stable for single neurons, can destabilize activity in otherwise stable recurrent networks leading to strong non-abating oscillations in the activity. This instability can be prevented by slowing down the homeostatic control. The stronger the network recurrence, the slower the homeostasis has to be. Next, we consider how non-linearities in the neural activation function affect these constraints. Finally, we consider the case that homeostatic feedback is mediated via a cascade of multiple intermediate stages. Counter-intuitively, the addition of extra stages in the homeostatic control loop further destabilizes activity in single neurons and networks. Our theoretical framework for homeostasis thus reveals previously unconsidered constraints on homeostasis in biological networks, and identifies conditions that require the slow time-constants of homeostatic regulation observed experimentally.

No MeSH data available.


Related in: MedlinePlus