Passive dendrites enable single neurons to compute linearly non-separable functions.
Passive dendrites enable single neurons to compute linearly non-separable functions.
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DOI：
10.1371/journal.pcbi.1002867
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发表时间：
2013
影响因子：
4.3
通讯作者：
Gutkin B
中科院分区：
文献类型：
--
作者：
Cazé RD;Humphries M;Gutkin B
锥体细胞树突中兴奋性输入的局部超线性求和，即所谓的树突棘波，导致独立的尖峰树突亚单位，将锥体神经元转变为能够计算线性不可分函数的两层神经网络，例如异或。其他神经元类，如中间神经元，可能只有几个独立的树突亚单位，或者只有被动树突，其中输入总和是纯粹的亚线性，并且树突亚单位只是饱和的。为了确定这种神经元是否也可以计算线性不可分函数，我们列举了对于给定的参数范围，可以由具有线性子单元和单个尖峰或饱和树枝状子单元的二元神经元模型实现的布尔函数。然后，我们将这些数值结果解析地推广到任意数量的非线性子单元。首先，我们证明了除了体细胞的非线性外，一个单一的非线性树枝状亚单位足以计算线性不可分函数。其次，我们解析地证明了，在有足够数量的饱和树突子单元的情况下，一个神经元可以计算纯兴奋输入下所有可计算的函数。第三，我们证明了这些线性不可分离的功能可以通过至少两种策略来实现：一种策略是树突亚单位足以触发体细胞棘波；另一种策略是体细胞棘波需要多个树突亚单位的合作。我们正式证明，后一种体系结构可以使用这两种类型的树枝状子单元来实现，而前者只可能使用尖峰树枝状结构。最后，我们展示了一个普通的两室生物物理模型和一个真实的小脑星状细胞中间神经元的神经元模型如何计算线性不可分函数。综上所述，我们的结果表明，被动树突足以使神经元计算线性不可分函数。关于单神经元计算的经典观点认为树突仅仅是输入的收集器，它被转发到SOMA进行线性求和，如果它足够大，就会导致尖峰输出。这样的单个神经元模型只能计算线性可分离的输入输出函数，代表所有可能函数的一小部分。最近的实验发现，在某些锥体细胞中，兴奋性输入可以超线性地整合到树突分支中，将该分支转变为尖峰树突亚单位。含有许多树突子单位的神经元可以计算线性可分离函数和线性不可分离函数。然而，其他类型的神经元有树突，因为缺乏所需的电压门控通道而不会出现尖峰。然而，这些树突将兴奋性输入亚线性求和，将分支转变为饱和的亚单位。我们想测试最后一种类型的非线性求和是否足以让单个神经元计算线性不可分函数。利用布尔代数和生物物理模型的结合，我们证明了具有单个非线性树枝状亚单位的神经元，无论是尖峰还是饱和，都能够计算线性不可分函数。因此，原则上，任何具有树状树的神经元，即使是被动的，也可以计算出线性不可分函数。
Local supra-linear summation of excitatory inputs occurring in pyramidal cell dendrites, the so-called dendritic spikes, results in independent spiking dendritic sub-units, which turn pyramidal neurons into two-layer neural networks capable of computing linearly non-separable functions, such as the exclusive OR. Other neuron classes, such as interneurons, may possess only a few independent dendritic sub-units, or only passive dendrites where input summation is purely sub-linear, and where dendritic sub-units are only saturating. To determine if such neurons can also compute linearly non-separable functions, we enumerate, for a given parameter range, the Boolean functions implementable by a binary neuron model with a linear sub-unit and either a single spiking or a saturating dendritic sub-unit. We then analytically generalize these numerical results to an arbitrary number of non-linear sub-units. First, we show that a single non-linear dendritic sub-unit, in addition to the somatic non-linearity, is sufficient to compute linearly non-separable functions. Second, we analytically prove that, with a sufficient number of saturating dendritic sub-units, a neuron can compute all functions computable with purely excitatory inputs. Third, we show that these linearly non-separable functions can be implemented with at least two strategies: one where a dendritic sub-unit is sufficient to trigger a somatic spike; another where somatic spiking requires the cooperation of multiple dendritic sub-units. We formally prove that implementing the latter architecture is possible with both types of dendritic sub-units whereas the former is only possible with spiking dendrites. Finally, we show how linearly non-separable functions can be computed by a generic two-compartment biophysical model and a realistic neuron model of the cerebellar stellate cell interneuron. Taken together our results demonstrate that passive dendrites are sufficient to enable neurons to compute linearly non-separable functions. Classical views on single neuron computation treat dendrites as mere collectors of inputs, that is forwarded to the soma for linear summation and causes a spike output if it is sufficiently large. Such a single neuron model can only compute linearly separable input-output functions, representing a small fraction of all possible functions. Recent experimental findings show that in certain pyramidal cells excitatory inputs can be supra-linearly integrated within a dendritic branch, turning this branch into a spiking dendritic sub-unit. Neurons containing many of these dendritic sub-units can compute both linearly separable and linearly non-separable functions. Nevertheless, other neuron types have dendrites which do not spike because the required voltage gated channels are absent. However, these dendrites sub-linearly sum excitatory inputs turning branches into saturating sub-units. We wanted to test if this last type of non-linear summation is sufficient for a single neuron to compute linearly non-separable functions. Using a combination of Boolean algebra and biophysical modeling, we show that a neuron with a single non-linear dendritic sub-unit whether spiking or saturating is able to compute linearly non-separable functions. Thus, in principle, any neuron with a dendritic tree, even passive, can compute linearly non-separable functions.
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影响因子：
2.5
作者：
Jaffe, DB;Carnevale, NT
通讯作者：
Carnevale, NT
影响因子：
25
作者：
Archie, KA;Mel, BW
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Mel, BW
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作者：
Gurney, KN
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Gurney, KN
影响因子：
2.5
作者：
Heck, D;Borst, A;Antkowiak, B
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Antkowiak, B
影响因子：
2.5
作者：
BERNANDER, O;KOCH, C;DOUGLAS, RJ
通讯作者：
DOUGLAS, RJ