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Compartmental modelling of dendrites deals with
multi-compartment modelling of the
dendrites, to make the understanding of the electrical behavior of complex dendrites easier. Basically, compartmental modelling of dendrites is a very helpful tool to develop new
biological neuron models. Dendrites are very important because they occupy the most membrane area in many of the neurons and give the neuron an ability to connect to thousands of other cells. Originally the dendrites were thought to have constant
conductance and
current but now it has been understood that they may have active
Voltage-gated ion channels, which influences the firing properties of the neuron and also the response of
neuron to
synaptic inputs.[1] Many mathematical models have been developed to understand the electric behavior of the dendrites. Dendrites tend to be very branchy and complex, so the compartmental approach to understand the electrical behavior of the dendrites makes it very useful.
[1][2]
Introduction
Compartmental modeling is a very natural way of modeling dynamical systems that have certain inherent properties with conservation principles. The compartmental modeling is an elegant way, a state space formulation to elegantly capture the dynamical systems that are governed by the conservation laws. Whether it is the conservation of mass, energy, fluid flow or information flow. Basically, they are models whose
state variables tend to be
non-negative (such as mass, concentrations, energy). So the equations for mass balance, energy, concentration or fluid flow can be written. It ultimately goes down to networks in which the brain is the largest of them all, just like
Avogadro number, very large amount of molecules that are interconnected. The brain has very interesting interconnections. On a microscopic level
thermodynamics is virtually impossible to understand but from a
macroscopic view we see that these follow some universal laws. In the same way brain has numerous interconnections, which is almost impossible to write a
differential equation for. (These words were said in an interview by
Dr. Wassim Haddad )
General observations about how the brain functions can be made by looking at the first and second
thermodynamic laws, which are universal laws.
Brain is a very large-scale interconnected system; the
neurons have to somehow behave like the chemical reaction system, so, it has to somehow obey the chemical thermodynamic laws. This approach may lead to more generalized model of brain. (These words were said in an interview by
Dr. Wassim Haddad )
Multiple compartments
a) Branched dendrites viewed as cylinders for modelling. b) Simple model with three compartments
Complicated dendritic structures can be treated as multiple compartments interconnected. The dendrites are divided into small compartments and they are linked together as shown in the figure.[1]
It is assumed that the compartment is isopotential and spatially uniform in its properties. Membrane non-uniformity such as diameter changes, and voltage differences are occurred in between the compartments but not inside them.[1]
An example of a simple two-compartment model:
Consider a two-compartmental model with the compartments viewed as isopotential cylinders with radius and length .
is the membrane potential of ith compartment.
is the specific membrane capacitance.
is the specific membrane resistivity.
The total electrode current, assuming that the compartment has it, is given by .
The longitudinal resistance is given by .
Now according to the balance that should exist for each compartment, we can say
.....eq(1)
Where and are the capacitive and ionic currents per unit area of ith compartment membrane. i.e. they can be given by
and .....eq(2)
If we assume the resting potential is 0. Then to compute , we need total axial resistance. As the compartments are simply cylinders we can say
.....eq(3)
Using ohms law we can express current from ith to jth compartment as
and .....eq(4)
The coupling terms and are obtained by inverting eq(3) and dividing by surface area of interest.
So we get,
and
Finally,
is the surface area of the compartment i.
If we put all these together we get
.....eq(5)
If we use and then eq(5) will become
.....eq(6)
Now if we inject current in cell 1 only and each cylinder is identical then
Without loss in generality we can define
After some algebra we can show that
also
i.e. the input resistance decreases. For increment in the potential, coupled system current should be greater than that is required for uncoupled system. This is because the second compartment drains some current.
Now, we can get a general compartmental model for a treelike structure and the equations are
Increased computational accuracy in multi-compartmental cable models
Input at the center
Each dendridic section is subdivided into segments, which are typically seen as uniform circular cylinders or tapered circular cylinders. In the traditional compartmental model, point process location is determined only to an accuracy of half segment length. This will make the model solution particularly sensitive to segment boundaries. The accuracy of the traditional approach for this reason is
O(1/n) when a point current and synaptic input is present. Usually the trans-membrane current where the membrane potential is known is represented in the model at points, or nodes and is assumed to be at the center. The new approach partitions the effect of the input by distributing it to the boundaries of the segment. Hence any input is partitioned between the nodes at the proximal and distal boundaries of the segment. Therefore, this procedure makes sure that the solution obtained is not sensitive to small changes in location of these boundaries because it affects how the input is partitioned between the nodes. When these compartments are connected with continuous potentials and conservation of current at segment boundaries then a new compartmental model of a new mathematical form is obtained. This new approach also provides a model identical to the traditional model but an order more accurate. This model increases the accuracy and precision by an order of magnitude than that is achieved by point process input.[2]
Cable Theory
Dendrites and axons are considered as continuous (cable like) than series of compartments[1] .
There is an awesome article that explains the cable theory on a whole.
Click here
Some applications
Information processing
A
theoretical framework along with a technological platform are provided by
computational models to enhance the understanding of
nervous system functions. There was a lot of advancement in the
molecular and
biophysical mechanisms underlying the neuronal activity. The same kind of advances have to be made in understanding the structure-functional relationship and rules followed by the information processing.[3]
Previously a neuron used to be thought as a transistor. However, it is shown recently that morphology and ionic composition of different neurons provide the cell with enhanced computational capabilities. These abilities are far more advanced than those captured by a point neuron.[3]
Some findings:
Different outputs given by the individual apical
oblique dendrites of
CA1 pyramidal neurons are linearly combined in the cell body. The outputs that come from these dendrites actually behave like individual computational units that use
sigmoidal activation function to combine inputs.[3]
Considering the accuracy in prediction of different input patterns by a two-layer neural network, its assumed that a simple mathematical equation can used to describe the model. This allows the development of network models in which each neuron, instead of being modeled as a full blown compartmental cell, it is modeled as a simplified two layer neural network.[3]
The firing pattern of the cell might contain the temporal information about incoming signals. For example, the delay between the two simulated pathways.[3]
Single CA1 has a capability of encoding and transmitting
spatio-temporal information on the incoming signals to the recipient cell.[3]
Calcium-activated nonspecific cationic (
CAN) mechanism is needed for giving constant activity and the
synapticstimulation alone does not induce persistent activity using the increasing conductance of
NMDA mechanism. NMDA/
AMPA positively expands the range of persistent activity and negatively regulates the amount of CAN needed for constant activity.[3]
The dopaminergic neurons have a low irregular basal firing frequency in 1-8 Hz range in vivo in the
ventral tegmental area (VTA) and
substantia nigra pars compacta (SNc). This frequencies can dramatically increase in response to a cue predicting reward or unpredicted reward. The actions that preceded the reward are reinforced by this burst or phasic signal.[4]
The low safety factor for action potential generation gives a result of low maximal steady frequencies. The transient initial frequency in response to depolarizing pulse is controlled by rate of Ca2+ accumulation in distal dendrites.[4]
Results obtained from a mulch-compartmental model realistic with reconstructed morphology were similar. So, the salient contributions of the dendritic architecture have been captures by simpler model.[4]
Mode locking
There are many important applications in neuroscience for
Mode-locking response of excitable systems to periodic forcing. For example, The theta rhythm drives the spatially extended place cells in the
hippocampus to generate a code giving information about spatial location. The role of neuronal dentrites in generating the response to periodic current injection can be explored by using a compartmental model (with linear dynamics for each compartment) coupled to an active soma model that generates action potentials.[5]
The shape of the tongues is influenced by the presence of the quasi-active membrane.[5]
The windows in parameter space for chaotic behavior can be enlarged with the resonant dendritic membrane.[5]
The response of the spatially extended neuron model to global forcing is different to that of point forcing.[5]
Compartmental neural simulations with spatial adaptivity
The computational cost of the meathod scales not with the physical size of the system being simulated but with the amount of activity present in the simulation. Spatial adaptivity for certain problems reduces up to 80%. [6]
Action potential (AP) initiation site
Establishing a unique site for AP initiation at the axon initial segment is no longer accepted. The APs can be initiated and comdected by different sub-regions of the neuron morphology, which widened the capabilities of individual neurons in computation.[7]
Findings from a study of the Action Potential Initiation Site Along the Axosomatodendritic Axis of Neurons Using Compartmental Models:
Dendritic APs are initiated more effectively by synchronous spatially clustered inputs than equivalent disperse inputs.[7]
The initiation site can also be determined by the average electrical distance from the dendritic input to the axon trigger zone, but it may be strongly modulated by the relative excitability of the two trigger zones and a number of factors.[7]
Using extracellular action potential recordings[9]
Using Multiple Voltage Recordings and Genetic Algorithms[10]
Multi-compartmental model of a CA1 pyramidal cell
Modelling reduced excitability in aged CA1 neurons as a calcium-dependent process.[11]
Electrical compartmentalization
Spine Neck Plasticity Controls Postsynaptic Calcium Signals through Electrical Compartmentalization.[12]
Robust coding in motion-sensitive neurons
Different receptive fields in axons and dendrites underlie robust coding in motion-sensitive neurons.[13]
Conductance-based neuron models
The capabilities and limitations of conductance-based compartmental
neuron models with reduced branched or unbranched morphologies and active
dendrites.[14]
Conclusion
Mathematics in neuroscience will influence it to a great degree, because most Neuroscientists have often relied on models based on intuition and words. On the other hand, if something is written that actually mathematically describes the process, and subject it to different kind of inputs and see the different kinds of effects (like observing
hallucinations, observing how
seizures might look like, or may be see how
consciousness and
unconsciousness might look like etc.). If we can start understanding some of these processes in neuroscience then we should be able to come up with much more general understanding based on these mathematical models than models that are just based on intuition or just observations between Neuroscientists, which only holds true for that one particular patient in that one particular moment.(These words were said in an interview by
Dr. Wassim Haddad )
^
abLindsay, A. E., Lindsay, K. A., & Rosenberg, J. R. (2005). Increased computational accuracy in multi-compartmental cable models by a novel approach for precise point process localization. Journal of Computational Neuroscience, 19(1), 21–38.
^
abcdefghPoirazi, P. (2009). Information Processing in Single Cells and Small Networks: Insights from Compartmental Models. In G. Maroulis & T. E. Simos (Eds.), Computational Methods in Science and Engineering, Vol 1 (Vol. 1108, pp. 158–167).
^
abcdKuznetsova, A. Y., Huertas, M. A., Kuznetsov, A. S., Paladini, C. A., & Canavier, C. C. (2010). Regulation of firing frequency in a computational model of a midbrain dopaminergic neuron. Journal of Computational Neuroscience, 28(3), 389–403.
^
abcdeSvensson, C. M., & Coombes, S. (2009). MODE LOCKING IN A SPATIALLY EXTENDED NEURON MODEL: ACTIVE SOMA AND COMPARTMENTAL TREE. International Journal of Bifurcation and Chaos, 19(8), 2597–2607.
^Rempe, M. J., Spruston, N., Kath, W. L., & Chopp, D. L. (2008). Compartmental neural simulations with spatial adaptivity. Journal of Computational Neuroscience, 25(3), 465–480.
^
abcIbarz, J. M., & Herreras, O. (2003). A study of the action potential initiation site along the axosomatodendritic axis of neurons using compartmental models. In J. Mira & J. R. Alvarez (Eds.), Computational Methods in Neural Modeling, Pt 1 (Vol. 2686, pp. 9–15).
^Schilstra, M., Rust, A., Adams, R., & Bolouri, H. (2002). A finite state automaton model for multi-neuron simulations. Neurocomputing, 44, 1141–1148.
^Gold, C., Henze, D. A., & Koch, C. (2007). Using extracellular action potential recordings to constrain compartmental models. Journal of Computational Neuroscience, 23(1), 39–58.
^Keren, N., Peled, N., & Korngreen, A. (2005). Constraining compartmental models using multiple voltage recordings and genetic algorithms. Journal of Neurophysiology, 94(6), 3730–3742.
^Markaki, M., Orphanoudakis, S., & Poirazi, P. (2005). Modelling reduced excitability in aged CA1 neurons as a calcium-dependent process. Neurocomputing, 65, 305–314.
^Grunditz, A., Holbro, N., Tian, L., Zuo, Y., & Oertner, T. G. (2008). Spine Neck Plasticity Controls Postsynaptic Calcium Signals through Electrical Compartmentalization. Journal of Neuroscience, 28(50), 13457–13466.
^Elyada, Y. M., Haag, J., & Borst, A. (2009). Different receptive fields in axons and dendrites underlie robust coding in motion-sensitive neurons. Nature Neuroscience, 12(3), 327–332.
^Hendrickson, E. B., Edgerton, J. R., & Jaeger, D. (2011). The capabilities and limitations of conductance-based compartmental neuron models with reduced branched or unbranched morphologies and active dendrites. Journal of Computational Neuroscience, 30(2), 301–321.
This is the user
sandbox of
Pradeepanandapu. A user sandbox is a subpage of the user's
user page. It serves as a testing spot and page development space for the user and is not an encyclopedia article. Create or edit your own sandbox
here.
Finished writing a draft article? Are you ready to request review of it by an experienced editor for possible inclusion in Wikipedia? Submit your draft for review!
Compartmental modelling of dendrites deals with
multi-compartment modelling of the
dendrites, to make the understanding of the electrical behavior of complex dendrites easier. Basically, compartmental modelling of dendrites is a very helpful tool to develop new
biological neuron models. Dendrites are very important because they occupy the most membrane area in many of the neurons and give the neuron an ability to connect to thousands of other cells. Originally the dendrites were thought to have constant
conductance and
current but now it has been understood that they may have active
Voltage-gated ion channels, which influences the firing properties of the neuron and also the response of
neuron to
synaptic inputs.[1] Many mathematical models have been developed to understand the electric behavior of the dendrites. Dendrites tend to be very branchy and complex, so the compartmental approach to understand the electrical behavior of the dendrites makes it very useful.
[1][2]
Introduction
Compartmental modeling is a very natural way of modeling dynamical systems that have certain inherent properties with conservation principles. The compartmental modeling is an elegant way, a state space formulation to elegantly capture the dynamical systems that are governed by the conservation laws. Whether it is the conservation of mass, energy, fluid flow or information flow. Basically, they are models whose
state variables tend to be
non-negative (such as mass, concentrations, energy). So the equations for mass balance, energy, concentration or fluid flow can be written. It ultimately goes down to networks in which the brain is the largest of them all, just like
Avogadro number, very large amount of molecules that are interconnected. The brain has very interesting interconnections. On a microscopic level
thermodynamics is virtually impossible to understand but from a
macroscopic view we see that these follow some universal laws. In the same way brain has numerous interconnections, which is almost impossible to write a
differential equation for. (These words were said in an interview by
Dr. Wassim Haddad )
General observations about how the brain functions can be made by looking at the first and second
thermodynamic laws, which are universal laws.
Brain is a very large-scale interconnected system; the
neurons have to somehow behave like the chemical reaction system, so, it has to somehow obey the chemical thermodynamic laws. This approach may lead to more generalized model of brain. (These words were said in an interview by
Dr. Wassim Haddad )
Multiple compartments
a) Branched dendrites viewed as cylinders for modelling. b) Simple model with three compartments
Complicated dendritic structures can be treated as multiple compartments interconnected. The dendrites are divided into small compartments and they are linked together as shown in the figure.[1]
It is assumed that the compartment is isopotential and spatially uniform in its properties. Membrane non-uniformity such as diameter changes, and voltage differences are occurred in between the compartments but not inside them.[1]
An example of a simple two-compartment model:
Consider a two-compartmental model with the compartments viewed as isopotential cylinders with radius and length .
is the membrane potential of ith compartment.
is the specific membrane capacitance.
is the specific membrane resistivity.
The total electrode current, assuming that the compartment has it, is given by .
The longitudinal resistance is given by .
Now according to the balance that should exist for each compartment, we can say
.....eq(1)
Where and are the capacitive and ionic currents per unit area of ith compartment membrane. i.e. they can be given by
and .....eq(2)
If we assume the resting potential is 0. Then to compute , we need total axial resistance. As the compartments are simply cylinders we can say
.....eq(3)
Using ohms law we can express current from ith to jth compartment as
and .....eq(4)
The coupling terms and are obtained by inverting eq(3) and dividing by surface area of interest.
So we get,
and
Finally,
is the surface area of the compartment i.
If we put all these together we get
.....eq(5)
If we use and then eq(5) will become
.....eq(6)
Now if we inject current in cell 1 only and each cylinder is identical then
Without loss in generality we can define
After some algebra we can show that
also
i.e. the input resistance decreases. For increment in the potential, coupled system current should be greater than that is required for uncoupled system. This is because the second compartment drains some current.
Now, we can get a general compartmental model for a treelike structure and the equations are
Increased computational accuracy in multi-compartmental cable models
Input at the center
Each dendridic section is subdivided into segments, which are typically seen as uniform circular cylinders or tapered circular cylinders. In the traditional compartmental model, point process location is determined only to an accuracy of half segment length. This will make the model solution particularly sensitive to segment boundaries. The accuracy of the traditional approach for this reason is
O(1/n) when a point current and synaptic input is present. Usually the trans-membrane current where the membrane potential is known is represented in the model at points, or nodes and is assumed to be at the center. The new approach partitions the effect of the input by distributing it to the boundaries of the segment. Hence any input is partitioned between the nodes at the proximal and distal boundaries of the segment. Therefore, this procedure makes sure that the solution obtained is not sensitive to small changes in location of these boundaries because it affects how the input is partitioned between the nodes. When these compartments are connected with continuous potentials and conservation of current at segment boundaries then a new compartmental model of a new mathematical form is obtained. This new approach also provides a model identical to the traditional model but an order more accurate. This model increases the accuracy and precision by an order of magnitude than that is achieved by point process input.[2]
Cable Theory
Dendrites and axons are considered as continuous (cable like) than series of compartments[1] .
There is an awesome article that explains the cable theory on a whole.
Click here
Some applications
Information processing
A
theoretical framework along with a technological platform are provided by
computational models to enhance the understanding of
nervous system functions. There was a lot of advancement in the
molecular and
biophysical mechanisms underlying the neuronal activity. The same kind of advances have to be made in understanding the structure-functional relationship and rules followed by the information processing.[3]
Previously a neuron used to be thought as a transistor. However, it is shown recently that morphology and ionic composition of different neurons provide the cell with enhanced computational capabilities. These abilities are far more advanced than those captured by a point neuron.[3]
Some findings:
Different outputs given by the individual apical
oblique dendrites of
CA1 pyramidal neurons are linearly combined in the cell body. The outputs that come from these dendrites actually behave like individual computational units that use
sigmoidal activation function to combine inputs.[3]
Considering the accuracy in prediction of different input patterns by a two-layer neural network, its assumed that a simple mathematical equation can used to describe the model. This allows the development of network models in which each neuron, instead of being modeled as a full blown compartmental cell, it is modeled as a simplified two layer neural network.[3]
The firing pattern of the cell might contain the temporal information about incoming signals. For example, the delay between the two simulated pathways.[3]
Single CA1 has a capability of encoding and transmitting
spatio-temporal information on the incoming signals to the recipient cell.[3]
Calcium-activated nonspecific cationic (
CAN) mechanism is needed for giving constant activity and the
synapticstimulation alone does not induce persistent activity using the increasing conductance of
NMDA mechanism. NMDA/
AMPA positively expands the range of persistent activity and negatively regulates the amount of CAN needed for constant activity.[3]
The dopaminergic neurons have a low irregular basal firing frequency in 1-8 Hz range in vivo in the
ventral tegmental area (VTA) and
substantia nigra pars compacta (SNc). This frequencies can dramatically increase in response to a cue predicting reward or unpredicted reward. The actions that preceded the reward are reinforced by this burst or phasic signal.[4]
The low safety factor for action potential generation gives a result of low maximal steady frequencies. The transient initial frequency in response to depolarizing pulse is controlled by rate of Ca2+ accumulation in distal dendrites.[4]
Results obtained from a mulch-compartmental model realistic with reconstructed morphology were similar. So, the salient contributions of the dendritic architecture have been captures by simpler model.[4]
Mode locking
There are many important applications in neuroscience for
Mode-locking response of excitable systems to periodic forcing. For example, The theta rhythm drives the spatially extended place cells in the
hippocampus to generate a code giving information about spatial location. The role of neuronal dentrites in generating the response to periodic current injection can be explored by using a compartmental model (with linear dynamics for each compartment) coupled to an active soma model that generates action potentials.[5]
The shape of the tongues is influenced by the presence of the quasi-active membrane.[5]
The windows in parameter space for chaotic behavior can be enlarged with the resonant dendritic membrane.[5]
The response of the spatially extended neuron model to global forcing is different to that of point forcing.[5]
Compartmental neural simulations with spatial adaptivity
The computational cost of the meathod scales not with the physical size of the system being simulated but with the amount of activity present in the simulation. Spatial adaptivity for certain problems reduces up to 80%. [6]
Action potential (AP) initiation site
Establishing a unique site for AP initiation at the axon initial segment is no longer accepted. The APs can be initiated and comdected by different sub-regions of the neuron morphology, which widened the capabilities of individual neurons in computation.[7]
Findings from a study of the Action Potential Initiation Site Along the Axosomatodendritic Axis of Neurons Using Compartmental Models:
Dendritic APs are initiated more effectively by synchronous spatially clustered inputs than equivalent disperse inputs.[7]
The initiation site can also be determined by the average electrical distance from the dendritic input to the axon trigger zone, but it may be strongly modulated by the relative excitability of the two trigger zones and a number of factors.[7]
Using extracellular action potential recordings[9]
Using Multiple Voltage Recordings and Genetic Algorithms[10]
Multi-compartmental model of a CA1 pyramidal cell
Modelling reduced excitability in aged CA1 neurons as a calcium-dependent process.[11]
Electrical compartmentalization
Spine Neck Plasticity Controls Postsynaptic Calcium Signals through Electrical Compartmentalization.[12]
Robust coding in motion-sensitive neurons
Different receptive fields in axons and dendrites underlie robust coding in motion-sensitive neurons.[13]
Conductance-based neuron models
The capabilities and limitations of conductance-based compartmental
neuron models with reduced branched or unbranched morphologies and active
dendrites.[14]
Conclusion
Mathematics in neuroscience will influence it to a great degree, because most Neuroscientists have often relied on models based on intuition and words. On the other hand, if something is written that actually mathematically describes the process, and subject it to different kind of inputs and see the different kinds of effects (like observing
hallucinations, observing how
seizures might look like, or may be see how
consciousness and
unconsciousness might look like etc.). If we can start understanding some of these processes in neuroscience then we should be able to come up with much more general understanding based on these mathematical models than models that are just based on intuition or just observations between Neuroscientists, which only holds true for that one particular patient in that one particular moment.(These words were said in an interview by
Dr. Wassim Haddad )
^
abLindsay, A. E., Lindsay, K. A., & Rosenberg, J. R. (2005). Increased computational accuracy in multi-compartmental cable models by a novel approach for precise point process localization. Journal of Computational Neuroscience, 19(1), 21–38.
^
abcdefghPoirazi, P. (2009). Information Processing in Single Cells and Small Networks: Insights from Compartmental Models. In G. Maroulis & T. E. Simos (Eds.), Computational Methods in Science and Engineering, Vol 1 (Vol. 1108, pp. 158–167).
^
abcdKuznetsova, A. Y., Huertas, M. A., Kuznetsov, A. S., Paladini, C. A., & Canavier, C. C. (2010). Regulation of firing frequency in a computational model of a midbrain dopaminergic neuron. Journal of Computational Neuroscience, 28(3), 389–403.
^
abcdeSvensson, C. M., & Coombes, S. (2009). MODE LOCKING IN A SPATIALLY EXTENDED NEURON MODEL: ACTIVE SOMA AND COMPARTMENTAL TREE. International Journal of Bifurcation and Chaos, 19(8), 2597–2607.
^Rempe, M. J., Spruston, N., Kath, W. L., & Chopp, D. L. (2008). Compartmental neural simulations with spatial adaptivity. Journal of Computational Neuroscience, 25(3), 465–480.
^
abcIbarz, J. M., & Herreras, O. (2003). A study of the action potential initiation site along the axosomatodendritic axis of neurons using compartmental models. In J. Mira & J. R. Alvarez (Eds.), Computational Methods in Neural Modeling, Pt 1 (Vol. 2686, pp. 9–15).
^Schilstra, M., Rust, A., Adams, R., & Bolouri, H. (2002). A finite state automaton model for multi-neuron simulations. Neurocomputing, 44, 1141–1148.
^Gold, C., Henze, D. A., & Koch, C. (2007). Using extracellular action potential recordings to constrain compartmental models. Journal of Computational Neuroscience, 23(1), 39–58.
^Keren, N., Peled, N., & Korngreen, A. (2005). Constraining compartmental models using multiple voltage recordings and genetic algorithms. Journal of Neurophysiology, 94(6), 3730–3742.
^Markaki, M., Orphanoudakis, S., & Poirazi, P. (2005). Modelling reduced excitability in aged CA1 neurons as a calcium-dependent process. Neurocomputing, 65, 305–314.
^Grunditz, A., Holbro, N., Tian, L., Zuo, Y., & Oertner, T. G. (2008). Spine Neck Plasticity Controls Postsynaptic Calcium Signals through Electrical Compartmentalization. Journal of Neuroscience, 28(50), 13457–13466.
^Elyada, Y. M., Haag, J., & Borst, A. (2009). Different receptive fields in axons and dendrites underlie robust coding in motion-sensitive neurons. Nature Neuroscience, 12(3), 327–332.
^Hendrickson, E. B., Edgerton, J. R., & Jaeger, D. (2011). The capabilities and limitations of conductance-based compartmental neuron models with reduced branched or unbranched morphologies and active dendrites. Journal of Computational Neuroscience, 30(2), 301–321.